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leandojo-lean4-formal-informal-strings-split

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MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁷ : Group G inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α c : G μ : Measure α inst✝³ : SMulInvariantMeasure G α μ inst✝² : TopologicalSpace α inst✝¹ : ContinuousConstSMul G α inst✝ : MulAction.IsMinimal G α K U : Set α hK : IsCompact K hμK : ↑↑μ K ≠ 0 hU : IsOpen U hne : Set.Nonempty U t : Finset G ht : K ⊆ ⋃ g ∈ t, g • U hμU : ↑↑μ U = 0 x✝¹ : G x✝ : x✝¹ ∈ ↑t ⊢ ↑↑μ (x✝¹ • U) = 0 rwa [measure_smul] ** Qed
MeasureTheory.isLocallyFiniteMeasure_of_smulInvariant G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁷ : Group G inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α c : G μ : Measure α inst✝³ : SMulInvariantMeasure G α μ inst✝² : TopologicalSpace α inst✝¹ : ContinuousConstSMul G α inst✝ : MulAction.IsMinimal G α K U : Set α hU : IsOpen U hne : Set.Nonempty U hμU : ↑↑μ U ≠ ⊤ x : α g : G hg : g • x ∈ U ⊢ ↑↑μ ((fun x x_1 => x • x_1) g ⁻¹' U) ≠ ⊤ rwa [measure_preimage_smul] ** Qed
MeasureTheory.smul_ae_eq_self_of_mem_zpowers G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x y : G hs : x • s =ᶠ[ae μ] s hy : y ∈ Subgroup.zpowers x ⊢ y • s =ᶠ[ae μ] s obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp hy case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x ⊢ x ^ k • s =ᶠ[ae μ] s let e : α ≃ α := MulAction.toPermHom G α x case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x ⊢ x ^ k • s =ᶠ[ae μ] s have he : QuasiMeasurePreserving e μ μ := (measurePreserving_smul x μ).quasiMeasurePreserving case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e ⊢ x ^ k • s =ᶠ[ae μ] s have he' : QuasiMeasurePreserving e.symm μ μ := (measurePreserving_smul x⁻¹ μ).quasiMeasurePreserving case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e he' : QuasiMeasurePreserving ↑e.symm ⊢ x ^ k • s =ᶠ[ae μ] s have h := he.image_zpow_ae_eq he' k hs case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e he' : QuasiMeasurePreserving ↑e.symm h : ↑(e ^ k) '' s =ᶠ[ae μ] s ⊢ x ^ k • s =ᶠ[ae μ] s simp only [← MonoidHom.map_zpow] at h case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e he' : QuasiMeasurePreserving ↑e.symm h : (fun a => ↑(↑(MulAction.toPermHom G α) (x ^ k)) a) '' s =ᶠ[ae μ] s ⊢ x ^ k • s =ᶠ[ae μ] s simpa only [MulAction.toPermHom_apply, MulAction.toPerm_apply, image_smul] using h ** Qed
Nat.mem_rfind p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m ⊢ n ∈ rfind p let ⟨m, hm⟩ := dom_iff_mem.1 <\| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩ p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p ⊢ n ∈ rfind p rcases lt_trichotomy m n with (h \| h \| h) case inl p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p h : m < n ⊢ n ∈ rfind p injection mem_unique (h₂ h) (rfind_spec hm) case inr.inl p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p h : m = n ⊢ n ∈ rfind p rwa [← h] case inr.inr p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p h : n < m ⊢ n ∈ rfind p injection mem_unique h₁ (rfind_min hm h) ** Qed
Nat.rfind_min' p : ℕ → Bool m : ℕ pm : p m = true this : true ∈ ↑p m n : ℕ hn : n ∈ rfind ↑p h : m < n ⊢ False injection mem_unique this (rfind_min hn h) ** Qed
Nat.rfindOpt_dom α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n ⊢ (rfindOpt f).Dom have h' : ∃ n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2 α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true ⊢ (rfindOpt f).Dom have s := Nat.find_spec h' α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true ⊢ (rfindOpt f).Dom have fd : (rfind fun n => (f n).isSome).Dom := ⟨Nat.find h', by simpa using s.symm, fun _ _ => trivial⟩ α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom ⊢ (rfindOpt f).Dom refine' ⟨fd, _⟩ α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom ⊢ ((fun b => (fun n => ↑(f n)) (Part.get (rfind fun n => ↑(Option.some (Option.isSome (f n)))) b)) fd).Dom have := rfind_spec (get_mem fd) α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true ∈ ↑(Option.some (Option.isSome (f (Part.get (rfind fun n => ↑(Option.some (Option.isSome (f n)))) fd)))) ⊢ ((fun b => (fun n => ↑(f n)) (Part.get (rfind fun n => ↑(Option.some (Option.isSome (f n)))) b)) fd).Dom simp at this ⊢ α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true = Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) ⊢ Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) = true cases' Option.isSome_iff_exists.1 this.symm with a e case intro α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true = Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) a : α e : f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)) = Option.some a ⊢ Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) = true rw [e] case intro α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true = Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) a : α e : f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)) = Option.some a ⊢ Option.isSome (Option.some a) = true trivial α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true ⊢ true ∈ (fun n => ↑(Option.some (Option.isSome (f n)))) (Nat.find h') simpa using s.symm ** Qed
Nat.rfindOpt_mono α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n ⊢ a ∈ rfindOpt f have h' := rfindOpt_dom.2 ⟨_, _, h⟩ α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom ⊢ a ∈ rfindOpt f cases' rfindOpt_spec ⟨h', rfl⟩ with k hk case intro α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom k : ℕ hk : Part.get (rfindOpt f) h' ∈ f k ⊢ a ∈ rfindOpt f have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk) case intro α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom k : ℕ hk : Part.get (rfindOpt f) h' ∈ f k this : Option.some a = Option.some (Part.get (rfindOpt f) h') ⊢ a ∈ rfindOpt f simp at this case intro α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom k : ℕ hk : Part.get (rfindOpt f) h' ∈ f k this : a = Part.get (rfindOpt f) h' ⊢ a ∈ rfindOpt f simp [this, get_mem] ** Qed
Nat.Partrec.of_primrec f : ℕ → ℕ hf : Nat.Primrec f ⊢ Partrec ↑f induction hf with \| zero => exact zero \| succ => exact succ \| left => exact left \| right => exact right \| pair _ _ pf pg => refine' (pf.pair pg).of_eq_tot fun n => _ simp [Seq.seq] \| comp _ _ pf pg => refine' (pf.comp pg).of_eq_tot fun n => _ simp \| prec _ _ pf pg => refine' (pf.prec pg).of_eq_tot fun n => _ simp only [unpaired, PFun.coe_val, bind_eq_bind] induction n.unpair.2 with \| zero => simp \| succ m IH => simp only [mem_bind_iff, mem_some_iff] exact ⟨_, IH, rfl⟩ case zero f : ℕ → ℕ ⊢ Partrec ↑fun x => 0 exact zero case succ f : ℕ → ℕ ⊢ Partrec ↑Nat.succ exact succ case left f : ℕ → ℕ ⊢ Partrec ↑fun n => (unpair n).1 exact left case right f : ℕ → ℕ ⊢ Partrec ↑fun n => (unpair n).2 exact right case pair f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ ⊢ Partrec ↑fun n => Nat.pair (f✝ n) (g✝ n) refine' (pf.pair pg).of_eq_tot fun n => _ case pair f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ Nat.pair (f✝ n) (g✝ n) ∈ Seq.seq (Nat.pair <$> ↑f✝ n) fun x => ↑g✝ n simp [Seq.seq] case comp f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ ⊢ Partrec ↑fun n => f✝ (g✝ n) refine' (pf.comp pg).of_eq_tot fun n => _ case comp f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ f✝ (g✝ n) ∈ ↑g✝ n >>= ↑f✝ simp case prec f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ ⊢ Partrec ↑(unpaired fun z n => Nat.rec (f✝ z) (fun y IH => g✝ (Nat.pair z (Nat.pair y IH))) n) refine' (pf.prec pg).of_eq_tot fun n => _ case prec f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ unpaired (fun z n => Nat.rec (f✝ z) (fun y IH => g✝ (Nat.pair z (Nat.pair y IH))) n) n ∈ unpaired (fun a n => Nat.rec (↑f✝ a) (fun y IH => do let i ← IH ↑g✝ (Nat.pair a (Nat.pair y i))) n) n simp only [unpaired, PFun.coe_val, bind_eq_bind] case prec f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) (unpair n).2 ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) (unpair n).2 induction n.unpair.2 with \| zero => simp \| succ m IH => simp only [mem_bind_iff, mem_some_iff] exact ⟨_, IH, rfl⟩ case prec.zero f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) Nat.zero ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) Nat.zero simp case prec.succ f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n m : ℕ IH : Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) m ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) m ⊢ Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) (Nat.succ m) ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) (Nat.succ m) simp only [mem_bind_iff, mem_some_iff] case prec.succ f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n m : ℕ IH : Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) m ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) m ⊢ ∃ a, a ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) m ∧ g✝ (Nat.pair (unpair n).1 (Nat.pair m (Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) m))) = g✝ (Nat.pair (unpair n).1 (Nat.pair m a)) exact ⟨_, IH, rfl⟩ ** Qed
Nat.Partrec.none n a : ℕ x✝ : a ∈ Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (↑fun x => 1) (Nat.pair n n_1) h : (Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (↑fun x => 1) (Nat.pair n n_1)).Dom h✝ : Part.get (Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (↑fun x => 1) (Nat.pair n n_1)) h = a ⊢ False simp at h ** Qed
Primrec.to_comp α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → σ hf : Primrec f n : ℕ ⊢ (do let n ← (↑fun n => encode (Option.map f (decode n))) n ↑(Nat.ppred n)) = Part.bind ↑(decode n) fun a => map encode (↑f a) simp α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → σ hf : Primrec f n : ℕ ⊢ ↑(Nat.ppred (encode (Option.map f (decode n)))) = Part.bind ↑(decode n) fun a => Part.some (encode (f a)) cases decode (α := α) n <;> simp ** Qed
Computable.ofOption α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β hf : Computable f n : ℕ ⊢ (do let n ← Part.bind ↑(decode n) fun a => map encode (↑f a) ↑(Nat.ppred n)) = Part.bind ↑(decode n) fun a => map encode ((fun a => ↑(f a)) a) cases' decode (α := α) n with a <;> simp case some α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β hf : Computable f n : ℕ a : α ⊢ ↑(Nat.ppred (encode (f a))) = map encode ↑(f a) cases' f a with b <;> simp ** Qed
Computable.pair α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β g : α → γ hf : Computable f hg : Computable g n : ℕ ⊢ (Seq.seq (Nat.pair <$> Part.bind ↑(decode n) fun a => map encode (↑f a)) fun x => Part.bind ↑(decode n) fun a => map encode (↑g a)) = Part.bind ↑(decode n) fun a => map encode ((↑fun a => (f a, g a)) a) cases decode (α := α) n <;> simp [Seq.seq] ** Qed
Partrec.none α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ ⊢ Part.none = Part.bind ↑(decode n) fun a => map encode ((fun x => Part.none) a) cases decode (α := α) n <;> simp ** Qed
Partrec.bind α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. β g : α → β →. σ hf : Partrec f hg : Partrec₂ g n : ℕ ⊢ (do let n ← Seq.seq (Nat.pair <$> Part.some n) fun x => Part.bind ↑(decode n) fun a => map encode (f a) Part.bind ↑(decode n) fun a => map encode ((fun p => g p.1 p.2) a)) = Part.bind ↑(decode n) fun a => map encode ((fun a => Part.bind (f a) (g a)) a) simp [Seq.seq] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. β g : α → β →. σ hf : Partrec f hg : Partrec₂ g n : ℕ ⊢ (Part.bind (Part.bind ↑(decode n) fun y => map (Nat.pair n) (map encode (f y))) fun n => Part.bind ↑(Option.bind (decode (Nat.unpair n).1) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).2)) fun a => map encode (g a.1 a.2)) = Part.bind ↑(decode n) fun a => Part.bind (f a) fun y => map encode (g a y) cases' e : decode (α := α) n with a <;> simp [e, encodek] ** Qed
Partrec.map α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. β g : α → β → σ hf : Partrec f hg : Computable₂ g ⊢ Partrec fun a => Part.map (g a) (f a) simpa [bind_some_eq_map] using @Partrec.bind _ _ _ _ _ _ _ (fun a => Part.some ∘ (g a)) hf hg ** Qed
Partrec.comp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : β →. σ g : α → β hf : Partrec f hg : Computable g n : ℕ ⊢ (do let n ← Part.bind ↑(decode n) fun a => Part.map encode (↑g a) Part.bind ↑(decode n) fun a => Part.map encode (f a)) = Part.bind ↑(decode n) fun a => Part.map encode ((fun a => f (g a)) a) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : β →. σ g : α → β hf : Partrec f hg : Computable g n : ℕ ⊢ (Part.bind (Part.bind ↑(decode n) fun a => Part.some (encode (g a))) fun n => Part.bind ↑(decode n) fun a => Part.map encode (f a)) = Part.bind ↑(decode n) fun a => Part.map encode (f (g a)) cases' e : decode (α := α) n with a <;> simp [e, encodek] ** Qed
Partrec.nat_iff α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : ℕ →. ℕ ⊢ Partrec f ↔ Nat.Partrec f simp [Partrec, map_id'] ** Qed
Partrec₂.unpaired α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 σ : Type u_5 inst✝⁴ : Primcodable α inst✝³ : Primcodable β inst✝² : Primcodable γ inst✝¹ : Primcodable δ inst✝ : Primcodable σ f : ℕ → ℕ →. α h : Partrec (Nat.unpaired f) ⊢ Partrec₂ f simpa using Partrec.comp (g := fun p : ℕ × ℕ => (p.1, p.2)) h Primrec₂.pair.to_comp ** Qed
Partrec.nat_casesOn_right α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α ⊢ Nat.rec (↑g a) (fun y IH => Part.bind IH fun i => h (a, y, i).1 (Nat.pred (f (a, y, i).1))) (f a) = Nat.casesOn (f a) (Part.some (g a)) (h a) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α ⊢ Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a (Nat.pred (f a))) (f a) = Nat.rec (Part.some (g a)) (fun n n_ih => h a n) (f a) cases' f a with n <;> simp case succ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ ⊢ (Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n) = h a n refine' ext fun b => ⟨fun H => _, fun H => _⟩ case succ.refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n ⊢ b ∈ h a n rcases mem_bind_iff.1 H with ⟨c, _, h₂⟩ case succ.refine'_1.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n c : σ left✝ : c ∈ Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n h₂ : b ∈ h a n ⊢ b ∈ h a n exact h₂ case succ.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n ⊢ b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n have : ∀ m, (Nat.rec (motive := fun _ => Part σ) (Part.some (g a)) (fun y IH => IH.bind fun _ => h a n) m).Dom := by intro m induction m <;> simp [, H.fst] * case succ.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n this : ∀ (m : ℕ), (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom ⊢ b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n exact ⟨⟨this n, H.fst⟩, H.snd⟩ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n ⊢ ∀ (m : ℕ), (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom intro m α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n m : ℕ ⊢ (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom ** induction m <;> simp [, H.fst] * Qed
Partrec.vector_mOfFn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α →. σ hf : ∀ (i : Fin (n + 1)), Partrec (f i) ⊢ Partrec fun a => Vector.mOfFn fun i => f i a simp only [Vector.mOfFn, Nat.add_eq, Nat.add_zero, pure_eq_some, bind_eq_bind] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α →. σ hf : ∀ (i : Fin (n + 1)), Partrec (f i) ⊢ Partrec fun a => Part.bind (f 0 a) fun a_1 => Part.bind (Vector.mOfFn fun i => f (Fin.succ i) a) fun v => Part.some (a_1 ::ᵥ v) exact (hf 0).bind (Partrec.bind ((vector_mOfFn fun i => hf i.succ).comp fst) (Primrec.vector_cons.to_comp.comp (snd.comp fst) snd)) ** Qed
Computable.bind_decode_iff α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ fun a n => Option.bind (decode n) (f a) n : ℕ ⊢ (Part.bind ↑(decode n) fun a => map encode ((fun a => Part.bind (do let n ← (↑fun n => encode (decode n)) a.2 ↑(Nat.ppred n)) ↑fun b => (fun a n => Option.bind (decode n) (f a)) (a, b).1.1 (a, b).1.2) a)) = Part.bind ↑(decode n) fun a => map encode ((↑fun p => f p.1 p.2) a) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ fun a n => Option.bind (decode n) (f a) n : ℕ ⊢ (Part.bind ↑(Option.bind (decode (Nat.unpair n).1) fun a => Option.some (a, (Nat.unpair n).2)) fun a => Part.bind ↑(Nat.ppred (encode (decode a.2))) fun y => Part.some (encode (Option.bind (decode a.2) (f a.1)))) = Part.bind ↑(Option.bind (decode (Nat.unpair n).1) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).2)) fun a => Part.some (encode (f a.1 a.2)) cases decode (α := α) n.unpair.1 <;> simp case some α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ fun a n => Option.bind (decode n) (f a) n : ℕ val✝ : α ⊢ (Part.bind ↑(Nat.ppred (encode (decode (Nat.unpair n).2))) fun y => Part.some (encode (Option.bind (decode (Nat.unpair n).2) (f val✝)))) = Part.bind ↑(Option.map (Prod.mk val✝) (decode (Nat.unpair n).2)) fun a => Part.some (encode (f a.1 a.2)) cases decode (α := β) n.unpair.2 <;> simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f ⊢ Computable₂ fun a n => Option.bind (decode n) (f a) have : Partrec fun a : α × ℕ => (encode (decode (α := β) a.2)).casesOn (some Option.none) fun n => Part.map (f a.1) (decode (α := β) n) := Partrec.nat_casesOn_right (h := fun (a : α × ℕ) (n : ℕ) ↦ map (fun b ↦ f a.1 b) (Part.ofOption (decode n))) (Primrec.encdec.to_comp.comp snd) (const Option.none) ((ofOption (Computable.decode.comp snd)).map (hf.comp (fst.comp <\| fst.comp fst) snd).to₂) α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f this : Partrec fun a => Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n) ⊢ Computable₂ fun a n => Option.bind (decode n) (f a) refine' this.of_eq fun a => _ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f this : Partrec fun a => Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n) a : α × ℕ ⊢ (Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n)) = (↑fun p => (fun a n => Option.bind (decode n) (f a)) p.1 p.2) a simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f this : Partrec fun a => Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n) a : α × ℕ ⊢ Nat.rec (Part.some Option.none) (fun n n_ih => map (f a.1) ↑(decode n)) (encode (decode a.2)) = Part.some (Option.bind (decode a.2) (f a.1)) cases decode (α := β) a.2 <;> simp [encodek] ** Qed
Computable.nat_rec α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ × σ → σ hf : Computable f hg : Computable g hh : Computable₂ h a : α ⊢ Nat.rec (↑g a) (fun y IH => Part.bind IH fun i => ↑(h a) (y, i)) (f a) = (↑fun a => Nat.rec (g a) (fun y IH => h a (y, IH)) (f a)) a simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ × σ → σ hf : Computable f hg : Computable g hh : Computable₂ h a : α ⊢ Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => Part.some (h a (y, i))) (f a) = Part.some (Nat.rec (g a) (fun y IH => h a (y, IH)) (f a)) induction f a <;> simp [] * Qed
Computable.cond α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ c : α → Bool f g : α → σ hc : Computable c hf : Computable f hg : Computable g a : α ⊢ (Nat.casesOn (encode (c a)) (g a) fun b => f (a, b).1) = bif c a then f a else g a cases c a <;> rfl ** Qed
Computable.option_bind α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β → Option σ hf : Computable f hg : Computable₂ g a : α ⊢ Option.casesOn (f a) Option.none (g a) = Option.bind (f a) (g a) cases f a <;> rfl ** Qed
Computable.option_map α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β → σ hf : Computable f hg : Computable₂ g ⊢ Computable fun a => Option.map (g a) (f a) convert option_bind hf (option_some.comp₂ hg) case h.e'_5.h α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β → σ hf : Computable f hg : Computable₂ g x✝ : α ⊢ Option.map (g x✝) (f x✝) = Option.bind (f x✝) fun b => Option.some (g x✝ b) apply Option.map_eq_bind ** Qed
Computable.option_getD α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β hf : Computable f hg : Computable g a : α ⊢ (Option.casesOn (f a) (g a) fun b => b) = Option.getD (f a) (g a) cases f a <;> rfl ** Qed
Computable.sum_casesOn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β ⊕ γ g : α → β → σ h : α → γ → σ hf : Computable f hg : Computable₂ g hh : Computable₂ h a : α ⊢ (bif Nat.bodd (encode (f a)) then Option.map (h a) (decode (Nat.div2 (encode (f a)))) else Option.map (g a) (decode (Nat.div2 (encode (f a))))) = Option.some (Sum.casesOn (f a) (g a) (h a)) cases' f a with b c <;> simp [Nat.div2_val] ** Qed
Computable.nat_strong_rec α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) this : Computable₂ fun a n => List.map (f a) (List.range n) a : α × ℕ ⊢ (fun a b => List.get? (List.map (f (a, b).1) (List.range (Nat.succ (a, b).2))) (a, b).2) a.1 a.2 = Option.some (f a.1 a.2) simp [List.get?_range (Nat.lt_succ_self a.2)] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind (a, y, IH).2.2 fun b => Option.map (fun b_1 => (((a, y, IH), b), b_1).1.2 ++ [(((a, y, IH), b), b_1).2]) (g ((a, y, IH), b).1.1.1 ((a, y, IH), b).2)) a.2 = Option.some ((fun a n => List.map (f a) (List.range n)) a.1 a.2) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) a.2 = Option.some (List.map (f a.1) (List.range a.2)) induction' a.2 with n IH case succ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ n : ℕ IH : Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) n = Option.some (List.map (f a.1) (List.range n)) ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) (Nat.succ n) = Option.some (List.map (f a.1) (List.range (Nat.succ n))) simp [IH, H, List.range_succ] case zero α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) Nat.zero = Option.some (List.map (f a.1) (List.range Nat.zero)) rfl ** Qed
Computable.list_ofFn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α → σ hf : ∀ (i : Fin (n + 1)), Computable (f i) ⊢ Computable fun a => List.ofFn fun i => f i a simp only [List.ofFn_succ] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α → σ hf : ∀ (i : Fin (n + 1)), Computable (f i) ⊢ Computable fun a => f 0 a :: List.ofFn fun i => f (Fin.succ i) a exact list_cons.comp (hf 0) (list_ofFn fun i => hf i.succ) ** Qed
Computable.vector_ofFn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin n → α → σ hf : ∀ (i : Fin n), Computable (f i) a : α ⊢ (Vector.mOfFn fun i => ↑(f i) a) = (↑fun a => Vector.ofFn fun i => f i a) a simp ** Qed
Partrec.option_some_iff α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. σ h : Partrec fun a => Part.map Option.some (f a) n : ℕ ⊢ (do let n ← Part.bind ↑(decode n) fun a => Part.map encode ((fun a => Part.map Option.some (f a)) a) ↑(Nat.ppred n)) = Part.bind ↑(decode n) fun a => Part.map encode (f a) simp [Part.bind_assoc, ← Function.comp_apply (f := Part.some) (g := encode), bind_some_eq_map, -Function.comp_apply] ** Qed
Partrec.option_casesOn_right α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ o : α → Option β f : α → σ g : α → β →. σ ho : Computable o hf : Computable f hg : Partrec₂ g this : Partrec fun a => Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (↑(decode n)) (g a) a : α ⊢ (Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (↑(decode n)) (g a)) = Option.casesOn (o a) (Part.some (f a)) (g a) cases' o a with b <;> simp [encodek] ** Qed
Partrec.sum_casesOn_left α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β ⊕ γ g : α → β →. σ h : α → γ → σ hf : Computable f hg : Partrec₂ g hh : Computable₂ h a : α ⊢ Sum.casesOn (Sum.casesOn (f a) (fun b => Sum.inr (a, b).2) fun b => Sum.inl (a, b).2) (fun b => Part.some (h a b)) (g a) = Sum.casesOn (f a) (g a) fun c => Part.some (h a c) cases f a <;> simp ** Qed
Partrec.fix_aux α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ ⊢ let F := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n; (∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n) ↔ b ∈ PFun.fix f a intro F α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n ⊢ (∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n) ↔ b ∈ PFun.fix f a refine' ⟨fun h => _, fun h => _⟩ case refine'_1 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n ⊢ b ∈ PFun.fix f a rcases h with ⟨n, ⟨_x, h₁⟩, h₂⟩ case refine'_1.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a ⊢ b ∈ PFun.fix f a cases n <;> simp at h₂ case refine'_1.intro.intro.intro.succ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m h₂ : Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ ∨ ∃ b_1, Sum.inr b_1 ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ ∧ Sum.inl b ∈ f b_1 ⊢ b ∈ PFun.fix f a rcases h₂ with (h₂ \| ⟨a', am', fa'⟩) α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m ⊢ ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a intro m a' am ba α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m : ℕ a' : α am : Sum.inr a' ∈ F a m ba : b ∈ PFun.fix f a' ⊢ b ∈ PFun.fix f a induction' m with m IH generalizing a' <;> simp at am case succ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m✝ : ℕ a'✝ : α am✝ : Sum.inr a'✝ ∈ F a m✝ ba✝ : b ∈ PFun.fix f a'✝ m : ℕ IH : ∀ (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a a' : α ba : b ∈ PFun.fix f a' am : ∃ b, Sum.inr b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m ∧ Sum.inr a' ∈ f b ⊢ b ∈ PFun.fix f a rcases am with ⟨a₂, am₂, fa₂⟩ case succ.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m✝ : ℕ a'✝ : α am : Sum.inr a'✝ ∈ F a m✝ ba✝ : b ∈ PFun.fix f a'✝ m : ℕ IH : ∀ (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a a' : α ba : b ∈ PFun.fix f a' a₂ : α am₂ : Sum.inr a₂ ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m fa₂ : Sum.inr a' ∈ f a₂ ⊢ b ∈ PFun.fix f a exact IH _ am₂ (PFun.mem_fix_iff.2 (Or.inr ⟨_, fa₂, ba⟩)) case zero α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m : ℕ a'✝ : α am✝ : Sum.inr a'✝ ∈ F a m ba✝ : b ∈ PFun.fix f a'✝ a' : α ba : b ∈ PFun.fix f a' am : a' = a ⊢ b ∈ PFun.fix f a rwa [← am] case refine'_1.intro.intro.intro.succ.inl α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m h₂ : Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ ⊢ b ∈ PFun.fix f a cases' h₁ (Nat.lt_succ_self _) with a' h case refine'_1.intro.intro.intro.succ.inl.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m h₂ : Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ a' : α h : Sum.inr a' ∈ F a n✝ ⊢ b ∈ PFun.fix f a injection mem_unique h h₂ case refine'_1.intro.intro.intro.succ.inr.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m a' : α am' : Sum.inr a' ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ fa' : Sum.inl b ∈ f a' ⊢ b ∈ PFun.fix f a exact this _ _ am' (PFun.mem_fix_iff.2 (Or.inl fa')) case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a ⊢ ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n suffices ∀ (a') (_ : b ∈ PFun.fix f a') (k) (_ : Sum.inr a' ∈ F a k), ∃ n, Sum.inl b ∈ F a n ∧ ∀ m < n, ∀ (_ : k ≤ m), ∃ a₂, Sum.inr a₂ ∈ F a m by rcases this _ h 0 (by simp) with ⟨n, hn₁, hn₂⟩ exact ⟨_, ⟨⟨_, hn₁⟩, fun {m} mn => hn₂ m mn (Nat.zero_le _)⟩, hn₁⟩ case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a ⊢ ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m intro a₁ h₁ case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ ⊢ ∀ (k : ℕ), Sum.inr a₁ ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m apply @PFun.fixInduction _ _ _ _ _ _ h₁ case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ ⊢ ∀ (a' : α), b ∈ PFun.fix f a' → (∀ (a'' : α), Sum.inr a'' ∈ f a' → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m) → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m intro a₂ h₂ IH k hk case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m rcases PFun.mem_fix_iff.1 h₂ with (h₂ \| ⟨a₃, am₃, _⟩) α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a this : ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n rcases this _ h 0 (by simp) with ⟨n, hn₁, hn₂⟩ case intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a this : ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → 0 ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n exact ⟨_, ⟨⟨_, hn₁⟩, fun {m} mn => hn₂ m mn (Nat.zero_le _)⟩, hn₁⟩ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a this : ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ Sum.inr a ∈ F a 0 simp case refine'_2.inl α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m refine' ⟨k.succ, _, fun m mk km => ⟨a₂, _⟩⟩ case refine'_2.inl.refine'_1 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ ⊢ Sum.inl b ∈ F a (Nat.succ k) simp case refine'_2.inl.refine'_1 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ ⊢ Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k ∨ ∃ b_1, Sum.inr b_1 ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k ∧ Sum.inl b ∈ f b_1 exact Or.inr ⟨_, hk, h₂⟩ case refine'_2.inl.refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ m : ℕ mk : m < Nat.succ k km : k ≤ m ⊢ Sum.inr a₂ ∈ F a m rwa [le_antisymm (Nat.le_of_lt_succ mk) km] case refine'_2.inr.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m rcases IH _ am₃ k.succ (by simp; exact ⟨_, hk, am₃⟩) with ⟨n, hn₁, hn₂⟩ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ ⊢ Sum.inr a₃ ∈ F a (Nat.succ k) simp α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ ⊢ ∃ b, Sum.inr b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k ∧ Sum.inr a₃ ∈ f b exact ⟨_, hk, am₃⟩ case refine'_2.inr.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m refine' ⟨n, hn₁, fun m mn km => _⟩ case refine'_2.inr.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m m : ℕ mn : m < n km : k ≤ m ⊢ ∃ a₂, Sum.inr a₂ ∈ F a m cases' km.lt_or_eq_dec with km km case refine'_2.inr.intro.intro.intro.intro.inl α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m m : ℕ mn : m < n km✝ : k ≤ m km : k < m ⊢ ∃ a₂, Sum.inr a₂ ∈ F a m exact hn₂ _ mn km case refine'_2.inr.intro.intro.intro.intro.inr α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m m : ℕ mn : m < n km✝ : k ≤ m km : k = m ⊢ ∃ a₂, Sum.inr a₂ ∈ F a m exact km ▸ ⟨_, hk⟩ ** Qed
ProbabilityTheory.kernel.IndepSets.symm α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α s₁ s₂ : Set (Set Ω) h : IndepSets s₁ s₂ κ ⊢ IndepSets s₂ s₁ κ intros t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α s₁ s₂ : Set (Set Ω) h : IndepSets s₁ s₂ κ t1 t2 : Set Ω ht1 : t1 ∈ s₂ ht2 : t2 ∈ s₁ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 filter_upwards [h t2 t1 ht2 ht1] with a ha case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α s₁ s₂ : Set (Set Ω) h : IndepSets s₁ s₂ κ t1 t2 : Set Ω ht1 : t1 ∈ s₂ ht2 : t2 ∈ s₁ a : α ha : ↑↑(↑κ a) (t2 ∩ t1) = ↑↑(↑κ a) t2 * ↑↑(↑κ a) t1 ⊢ ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rwa [Set.inter_comm, mul_comm] Qed
ProbabilityTheory.kernel.indep_bot_right α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ ⊢ Indep m' ⊥ κ intros s t _ ht ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} ht : t ∈ {s \| MeasurableSet s} ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} ht : t = ∅ ∨ t = Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t refine Filter.eventually_of_forall (fun a ↦ ?_) α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} ht : t = ∅ ∨ t = Set.univ a : α ⊢ ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t cases' ht with ht ht case inl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} a : α ht : t = ∅ ⊢ ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t rw [ht, Set.inter_empty, measure_empty, mul_zero] case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} a : α ht : t = Set.univ ⊢ ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t rw [ht, Set.inter_univ, measure_univ, mul_one] Qed
ProbabilityTheory.kernel.indepSet_empty_right α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : Set Ω ⊢ IndepSet s ∅ κ simp only [IndepSet, generateFrom_singleton_empty] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : Set Ω ⊢ Indep (generateFrom {s}) ⊥ κ exact indep_bot_right _ ** Qed
ProbabilityTheory.kernel.IndepSets.union α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ ⊢ IndepSets (s₁ ∪ s₂) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ t1 t2 : Set Ω ht1 : t1 ∈ s₁ ∪ s₂ ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 cases' (Set.mem_union _ _ _).mp ht1 with ht1₁ ht1₂ case inl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ t1 t2 : Set Ω ht1 : t1 ∈ s₁ ∪ s₂ ht2 : t2 ∈ s' ht1₁ : t1 ∈ s₁ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h₁ t1 t2 ht1₁ ht2 case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ t1 t2 : Set Ω ht1 : t1 ∈ s₁ ∪ s₂ ht2 : t2 ∈ s' ht1₂ : t1 ∈ s₂ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h₂ t1 t2 ht1₂ ht2 Qed
ProbabilityTheory.kernel.IndepSets.iUnion α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ ⊢ IndepSets (⋃ n, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋃ n, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rw [Set.mem_iUnion] at ht1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : ∃ i, t1 ∈ s i ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 cases' ht1 with n ht1 case intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ t1 t2 : Set Ω ht2 : t2 ∈ s' n : ι ht1 : t1 ∈ s n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact hyp n t1 t2 ht1 ht2 Qed
ProbabilityTheory.kernel.IndepSets.bUnion α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ ⊢ IndepSets (⋃ n ∈ u, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋃ n ∈ u, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 simp_rw [Set.mem_iUnion] at ht1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ t1 t2 : Set Ω ht2 : t2 ∈ s' ht1 : ∃ i i_1, t1 ∈ s i ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rcases ht1 with ⟨n, hpn, ht1⟩ case intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ t1 t2 : Set Ω ht2 : t2 ∈ s' n : ι hpn : n ∈ u ht1 : t1 ∈ s n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact hyp n hpn t1 t2 ht1 ht2 Qed
ProbabilityTheory.kernel.IndepSets.iInter α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h : ∃ n, IndepSets (s n) s' κ ⊢ IndepSets (⋂ n, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h : ∃ n, IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 cases' h with n h case intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n, s n ht2 : t2 ∈ s' n : ι h : IndepSets (s n) s' κ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h t1 t2 (Set.mem_iInter.mp ht1 n) ht2 Qed
ProbabilityTheory.kernel.IndepSets.bInter α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι h : ∃ n, n ∈ u ∧ IndepSets (s n) s' κ ⊢ IndepSets (⋂ n ∈ u, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι h : ∃ n, n ∈ u ∧ IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n ∈ u, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rcases h with ⟨n, hn, h⟩ case intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n ∈ u, s n ht2 : t2 ∈ s' n : ι hn : n ∈ u h : IndepSets (s n) s' κ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h t1 t2 (Set.biInter_subset_of_mem hn ht1) ht2 Qed
ProbabilityTheory.kernel.indepSets_singleton_iff ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s t : Set Ω _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t s1 t1 : Set Ω hs1 : s1 ∈ {s} ht1 : t1 ∈ {t} ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s1 ∩ t1) = ↑↑(↑κ a) s1 * ↑↑(↑κ a) t1 rwa [Set.mem_singleton_iff.mp hs1, Set.mem_singleton_iff.mp ht1] Qed
ProbabilityTheory.kernel.iIndep.indep α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m : ι → MeasurableSpace Ω _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : iIndep m κ i j : ι hij : i ≠ j ⊢ Indep (m i) (m j) κ change IndepSets ((fun x => MeasurableSet[m x]) i) ((fun x => MeasurableSet[m x]) j) κ μ α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m : ι → MeasurableSpace Ω _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : iIndep m κ i j : ι hij : i ≠ j ⊢ IndepSets ((fun x => MeasurableSet) i) ((fun x => MeasurableSet) j) κ exact iIndepSets.indepSets h_indep hij ** Qed
ProbabilityTheory.kernel.IndepSets.indep_aux ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 ** refine @induction_on_inter _ (fun t ↦ ∀ᵐ a ∂μ, κ a (t1 ∩ t) = κ a t1 * κ a t) _ m₂ hpm2 hp2 ?_ ?_ ?_ ?_ t2 ht2m ** case refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) ∅ simp only [Set.inter_empty, measure_empty, mul_zero, eq_self_iff_true, Filter.eventually_true] case refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ (t : Set Ω), t ∈ p2 → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) t exact fun t ht_mem_p2 ↦ hyp t1 t ht1 ht_mem_p2 case refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ (t : Set Ω), MeasurableSet t → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) t → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) tᶜ intros t ht h case refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ tᶜ) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) tᶜ filter_upwards [h] with a ha case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t a : α ha : ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t ⊢ ↑↑(↑κ a) (t1 ∩ tᶜ) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) tᶜ have : t1 ∩ tᶜ = t1 \ (t1 ∩ t) := by rw [Set.diff_self_inter, Set.diff_eq_compl_inter, Set.inter_comm] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t a : α ha : ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t this : t1 ∩ tᶜ = t1 \ (t1 ∩ t) ⊢ ↑↑(↑κ a) (t1 ∩ tᶜ) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) tᶜ rw [this, measure_diff (Set.inter_subset_left _ _) (ht1m.inter (h2 _ ht)) (measure_ne_top (κ a) _), measure_compl (h2 _ ht) (measure_ne_top (κ a) t), measure_univ, ENNReal.mul_sub (fun _ _ ↦ measure_ne_top (κ a) _), mul_one, ha] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t a : α ha : ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t ⊢ t1 ∩ tᶜ = t1 \ (t1 ∩ t) rw [Set.diff_self_inter, Set.diff_eq_compl_inter, Set.inter_comm] case refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ (f : ℕ → Set Ω), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) (f i)) → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) (⋃ i, f i) intros f hf_disj hf_meas h case refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ (i : ℕ), (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) (f i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ ⋃ i, f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) rw [← ae_all_iff] at h case refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ ⋃ i, f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) filter_upwards [h] with a ha case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ↑↑(↑κ a) (t1 ∩ ⋃ i, f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) rw [Set.inter_iUnion, measure_iUnion] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∑' (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) rw [measure_iUnion hf_disj (fun i ↦ h2 _ (hf_meas i))] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∑' (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ∑' (i : ℕ), ↑↑(↑κ a) (f i) rw [← ENNReal.tsum_mul_left] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∑' (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ∑' (i : ℕ), ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) congr with i case h.e_f.h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) i : ℕ ⊢ ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) rw [ha i] case h.hn α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ Pairwise (Disjoint on fun i => t1 ∩ f i) intros i j hij case h.hn α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) i j : ℕ hij : i ≠ j ⊢ (Disjoint on fun i => t1 ∩ f i) i j rw [Function.onFun, Set.inter_comm t1, Set.inter_comm t1] case h.hn α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) i j : ℕ hij : i ≠ j ⊢ Disjoint (f i ∩ t1) (f j ∩ t1) exact Disjoint.inter_left _ (Disjoint.inter_right _ (hf_disj hij)) case h.h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∀ (i : ℕ), MeasurableSet (t1 ∩ f i) exact fun i ↦ ht1m.inter (h2 _ (hf_meas i)) Qed
ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_of_disjoint α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ Indep (generateFrom {t \| ∃ n, n ∈ S ∧ s n = t}) (generateFrom {t \| ∃ k, k ∈ T ∧ s k = t}) κ rw [← generateFrom_piiUnionInter_singleton_left, ← generateFrom_piiUnionInter_singleton_left] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ Indep (generateFrom (piiUnionInter (fun k => {s k}) S)) (generateFrom (piiUnionInter (fun k => {s k}) T)) κ refine' IndepSets.indep' (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurableSet_generateFrom ht)) (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurableSet_generateFrom ht)) _ _ _ case refine'_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T t : Set Ω ht : t ∈ piiUnionInter (fun k => {s k}) S ⊢ ∀ (n : ι), generateFrom {s n} ≤ _mΩ exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k case refine'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T t : Set Ω ht : t ∈ piiUnionInter (fun k => {s k}) T ⊢ ∀ (n : ι), generateFrom {s n} ≤ _mΩ exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k case refine'_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun k => {s k}) S) exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _ case refine'_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun k => {s k}) T) exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _ case refine'_5 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun k => {s k}) S) (piiUnionInter (fun k => {s k}) T) κ classical exact indepSets_piiUnionInter_of_disjoint (iIndep.iIndepSets (fun n => rfl) hs) hST case refine'_5 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun k => {s k}) S) (piiUnionInter (fun k => {s k}) T) κ exact indepSets_piiUnionInter_of_disjoint (iIndep.iIndepSets (fun n => rfl) hs) hST ** Qed
ProbabilityTheory.kernel.indep_iSup_of_disjoint α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) κ refine' IndepSets.indep (iSup₂_le fun i _ => h_le i) (iSup₂_le fun i _ => h_le i) _ _ (generateFrom_piiUnionInter_measurableSet m S).symm (generateFrom_piiUnionInter_measurableSet m T).symm _ case refine'_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun n => {s \| MeasurableSet s}) S) exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _ case refine'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun n => {s \| MeasurableSet s}) T) exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _ case refine'_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun n => {s \| MeasurableSet s}) S) (piiUnionInter (fun n => {s \| MeasurableSet s}) T) κ classical exact indepSets_piiUnionInter_of_disjoint h_indep hST case refine'_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun n => {s \| MeasurableSet s}) S) (piiUnionInter (fun n => {s \| MeasurableSet s}) T) κ exact indepSets_piiUnionInter_of_disjoint h_indep hST ** Qed
ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_lt α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : Preorder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i : ι ⊢ Indep (generateFrom {s i}) (generateFrom {t \| ∃ j, j < i ∧ s j = t}) κ convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {i} { j \| j < i } (Set.disjoint_singleton_left.mpr (lt_irrefl _)) case h.e'_4.h.e'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : Preorder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i : ι ⊢ {s i} = {t \| ∃ n, n ∈ {i} ∧ s n = t} simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton'] ** Qed
ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_le α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : LinearOrder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i k : ι hk : i < k ⊢ Indep (generateFrom {s k}) (generateFrom {t \| ∃ j, j ≤ i ∧ s j = t}) κ convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {k} { j \| j ≤ i } (Set.disjoint_singleton_left.mpr hk.not_le) case h.e'_4.h.e'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : LinearOrder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i k : ι hk : i < k ⊢ {s k} = {t \| ∃ n, n ∈ {k} ∧ s n = t} simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton'] ** Qed
ProbabilityTheory.kernel.iIndepSets.piiUnionInter_of_not_mem α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S ⊢ IndepSets (piiUnionInter π ↑S) (π a) κ rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia ** case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : ↑s ⊆ ↑S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rw [Finset.coe_subset] at hs_mem case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ) case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS]) case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_μ_t1 : ∀ᵐ a' ∂μ, κ a' t1 = ∏ n in s, κ a' (f n) := by filter_upwards [hp_ind s h_f_mem_pi] with a' ha' rw [h_t1, ← ha'] case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_t2 : t2 = f a := by simp case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_μ_inter : ∀ᵐ a' ∂μ, κ a' (t1 ∩ t2) = ∏ n in insert a s, κ a' (f n) := by have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm] filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha' rw [h_t1_inter_t2, ← ha'] case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_μ_inter : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have has : a ∉ s := fun has_mem => haS (hs_mem has_mem) case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_μ_inter : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) has : ¬a ∈ s ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 filter_upwards [h_μ_t1, h_μ_inter] with a' ha1 ha2 case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_μ_inter : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) has : ¬a ∈ s a' : α ha1 : ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) ha2 : ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) ⊢ ↑↑(↑κ a') (t1 ∩ t2) = ↑↑(↑κ a') t1 * ↑↑(↑κ a') t2 rw [ha2, Finset.prod_insert has, h_t2, mul_comm, ha1] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ ⊢ ∀ (n : ι), n ∈ insert a s → f n ∈ π n intro n hn_mem_insert α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s ⊢ f n ∈ π n dsimp only α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem case inl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n = a ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n simp [hn_mem, ht2_mem_pia] case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n ∈ s ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem) case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n ∈ s hn_ne_a : n ≠ a ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n simp [hn_ne_a, hn_mem, hft1_mem n hn_mem] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n ∈ s ⊢ n ≠ a rintro rfl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) S : Finset ι hp_ind : iIndepSets π κ t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x n : ι hn_mem : n ∈ s haS : ¬n ∈ S ht2_mem_pia : t2 ∈ π n f : ι → Set Ω := fun n_1 => if n_1 = n then t2 else if n_1 ∈ s then ft1 n_1 else Set.univ hn_mem_insert : n ∈ insert n s ⊢ False exact haS (hs_mem hn_mem) α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n x : ι hxS : x ∈ s ⊢ x ∈ insert a s simp [hxS] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n ⊢ t1 = ⋂ n ∈ s, f n suffices h_forall : ∀ n ∈ s, f n = ft1 n case h_forall α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n ⊢ ∀ (n : ι), n ∈ s → f n = ft1 n intro n hnS case h_forall α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n n : ι hnS : n ∈ s ⊢ f n = ft1 n have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS) case h_forall α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n n : ι hnS : n ∈ s hn_ne_a : n ≠ a ⊢ f n = ft1 n simp_rw [if_pos hnS, if_neg hn_ne_a] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n ⊢ t1 = ⋂ n ∈ s, f n rw [ht1_eq] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n ⊢ ⋂ x ∈ s, ft1 x = ⋂ n ∈ s, f n ext x case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n x : Ω ⊢ x ∈ ⋂ x ∈ s, ft1 x ↔ x ∈ ⋂ n ∈ s, f n simp_rw [Set.mem_iInter] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n x : Ω ⊢ (∀ (i : ι), i ∈ s → x ∈ ft1 i) ↔ ∀ (i : ι), i ∈ s → x ∈ if i = a then t2 else if i ∈ s then ft1 i else Set.univ conv => lhs; intro i hns; rw [← h_forall i hns] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n n : ι hnS : n ∈ s ⊢ n ≠ a rintro rfl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) S : Finset ι hp_ind : iIndepSets π κ t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x n : ι hnS : n ∈ s haS : ¬n ∈ S ht2_mem_pia : t2 ∈ π n f : ι → Set Ω := fun n_1 => if n_1 = n then t2 else if n_1 ∈ s then ft1 n_1 else Set.univ h_f_mem : ∀ (n_1 : ι), n_1 ∈ insert n s → f n_1 ∈ π n_1 h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n ⊢ False exact haS (hs_mem hnS) α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n ⊢ ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) filter_upwards [hp_ind s h_f_mem_pi] with a' ha' case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n a' : α ha' : ↑↑(↑κ a') (⋂ i ∈ s, f i) = ∏ i in s, ↑↑(↑κ a') (f i) ⊢ ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) rw [h_t1, ← ha'] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) ⊢ t2 = f a simp α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a ⊢ ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n ⊢ ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha' case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n a' : α ha' : ↑↑(↑κ a') (⋂ i ∈ insert a s, f i) = ∏ i in insert a s, ↑↑(↑κ a') (f i) ⊢ ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) rw [h_t1_inter_t2, ← ha'] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a ⊢ t1 ∩ t2 = ⋂ n ∈ insert a s, f n rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm] Qed
ProbabilityTheory.kernel.indepSet_iff_indepSets_singleton α : Type u_1 Ω : Type u_2 ι : Type u_3 s t : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m0 : MeasurableSpace Ω hs_meas : MeasurableSet s ht_meas : MeasurableSet t κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ h : IndepSets {s} {t} κ u : Set Ω hu : u ∈ {s} ⊢ MeasurableSet u rwa [Set.mem_singleton_iff.mp hu] α : Type u_1 Ω : Type u_2 ι : Type u_3 s t : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m0 : MeasurableSpace Ω hs_meas : MeasurableSet s ht_meas : MeasurableSet t κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ h : IndepSets {s} {t} κ u : Set Ω hu : u ∈ {t} ⊢ MeasurableSet u rwa [Set.mem_singleton_iff.mp hu] ** Qed
ProbabilityTheory.kernel.Indep.indepSet_of_measurableSet α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t ⊢ IndepSet s t κ refine fun s' t' hs' ht' => h_indep s' t' ?_ ?_ case refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ s' ∈ {s \| MeasurableSet s} refine @generateFrom_induction _ (fun u => MeasurableSet[m₁] u) {s} ?_ ?_ ?_ ?_ _ hs' case refine_1.refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t : Set Ω), t ∈ {s} → (fun u => MeasurableSet u) t simp only [Set.mem_singleton_iff, forall_eq, hs] case refine_1.refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ (fun u => MeasurableSet u) ∅ exact @MeasurableSet.empty _ m₁ case refine_1.refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t : Set Ω), (fun u => MeasurableSet u) t → (fun u => MeasurableSet u) tᶜ exact fun u hu => hu.compl case refine_1.refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (f : ℕ → Set Ω), (∀ (n : ℕ), (fun u => MeasurableSet u) (f n)) → (fun u => MeasurableSet u) (⋃ i, f i) exact fun f hf => MeasurableSet.iUnion hf case refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ t' ∈ {s \| MeasurableSet s} refine @generateFrom_induction _ (fun u => MeasurableSet[m₂] u) {t} ?_ ?_ ?_ ?_ _ ht' case refine_2.refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t_1 : Set Ω), t_1 ∈ {t} → (fun u => MeasurableSet u) t_1 simp only [Set.mem_singleton_iff, forall_eq, ht] case refine_2.refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ (fun u => MeasurableSet u) ∅ exact @MeasurableSet.empty _ m₂ case refine_2.refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t : Set Ω), (fun u => MeasurableSet u) t → (fun u => MeasurableSet u) tᶜ exact fun u hu => hu.compl case refine_2.refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (f : ℕ → Set Ω), (∀ (n : ℕ), (fun u => MeasurableSet u) (f n)) → (fun u => MeasurableSet u) (⋃ i, f i) exact fun f hf => MeasurableSet.iUnion hf ** Qed
ProbabilityTheory.kernel.indepFun_iff_measure_inter_preimage_eq_mul ** α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' ⊢ IndepFun f g κ ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) constructor <;> intro h case mp α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' h : IndepFun f g κ ⊢ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) refine' fun s t hs ht => h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩ case mpr α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' h : ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) ⊢ IndepFun f g κ rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩ case mpr.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' h : ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) s : Set β hs : MeasurableSet s t : Set β' ht : MeasurableSet t ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) exact h s t hs ht Qed
ProbabilityTheory.kernel.iIndepFun_iff_measure_inter_preimage_eq_mul α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i ⊢ iIndepFun m f κ ↔ ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) refine' ⟨fun h S sets h_meas => h _ fun i hi_mem => ⟨sets i, h_meas i hi_mem, rfl⟩, _⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i ⊢ (∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i)) → iIndepFun m f κ intro h S setsΩ h_meas α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) let setsβ : ∀ i : ι, Set (β i) := fun i => dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).choose) fun _ => Set.univ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) have h_measβ : ∀ i ∈ S, MeasurableSet[m i] (setsβ i) := by intro i hi_mem simp_rw [dif_pos hi_mem] exact (h_meas i hi_mem).choose_spec.1 α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i := by intro i hi_mem simp_rw [dif_pos hi_mem] exact (h_meas i hi_mem).choose_spec.2.symm α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) have h_right_eq : ∀ a, (∏ i in S, κ a (setsΩ i)) = ∏ i in S, κ a ((f i) ⁻¹' (setsβ i)) := by refine' fun a ↦ Finset.prod_congr rfl fun i hi_mem => _ rw [h_preim i hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) h_right_eq : ∀ (a : α), ∏ i in S, ↑↑(↑κ a) (setsΩ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) filter_upwards [h S h_measβ] with a ha case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) h_right_eq : ∀ (a : α), ∏ i in S, ↑↑(↑κ a) (setsΩ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) a : α ha : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) ⊢ ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) rw [h_left_eq a, h_right_eq a, ha] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ ⊢ ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) intro i hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ i : ι hi_mem : i ∈ S ⊢ MeasurableSet (setsβ i) simp_rw [dif_pos hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ i : ι hi_mem : i ∈ S ⊢ MeasurableSet (Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i)) exact (h_meas i hi_mem).choose_spec.1 α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) ⊢ ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i intro i hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) i : ι hi_mem : i ∈ S ⊢ setsΩ i = f i ⁻¹' setsβ i simp_rw [dif_pos hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) i : ι hi_mem : i ∈ S ⊢ setsΩ i = f i ⁻¹' Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) exact (h_meas i hi_mem).choose_spec.2.symm α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i ⊢ ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) intro a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α ⊢ ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) congr with x case e_a.h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω ⊢ x ∈ ⋂ i ∈ S, setsΩ i ↔ x ∈ ⋂ i ∈ S, f i ⁻¹' setsβ i simp_rw [Set.mem_iInter] case e_a.h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω ⊢ (∀ (i : ι), i ∈ S → x ∈ setsΩ i) ↔ ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if h : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ constructor <;> intro h i hi_mem <;> specialize h i hi_mem case e_a.h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h✝ : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω i : ι hi_mem : i ∈ S h : x ∈ setsΩ i ⊢ x ∈ f i ⁻¹' if h : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ rwa [h_preim i hi_mem] at h case e_a.h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h✝ : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω i : ι hi_mem : i ∈ S h : x ∈ f i ⁻¹' if h : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ ⊢ x ∈ setsΩ i rwa [h_preim i hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) ⊢ ∀ (a : α), ∏ i in S, ↑↑(↑κ a) (setsΩ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) refine' fun a ↦ Finset.prod_congr rfl fun i hi_mem => _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) a : α i : ι hi_mem : i ∈ S ⊢ ↑↑(↑κ a) (setsΩ i) = ↑↑(↑κ a) (f i ⁻¹' setsβ i) rw [h_preim i hi_mem] ** Qed
ProbabilityTheory.kernel.indepFun_iff_indepSet_preimage α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g ⊢ IndepFun f g κ ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ refine' indepFun_iff_measure_inter_preimage_eq_mul.trans _ ** α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g ⊢ (∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t)) ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ constructor <;> intro h s t hs ht <;> specialize h s t hs ht case mp α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g s : Set β t : Set β' hs : MeasurableSet s ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) ⊢ IndepSet (f ⁻¹' s) (g ⁻¹' t) κ rwa [indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) κ μ] case mpr α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g s : Set β t : Set β' hs : MeasurableSet s ht : MeasurableSet t h : IndepSet (f ⁻¹' s) (g ⁻¹' t) κ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) rwa [← indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) κ μ] Qed
ProbabilityTheory.kernel.IndepFun.ae_eq α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' ⊢ IndepFun f' g' κ rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) filter_upwards [hf, hg, hfg _ _ ⟨_, hA, rfl⟩ ⟨_, hB, rfl⟩] with a hf' hg' hfg' case h α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B a : α hf' : f =ᵐ[↑κ a] f' hg' : g =ᵐ[↑κ a] g' hfg' : ↑↑(↑κ a) (f ⁻¹' A ∩ g ⁻¹' B) = ↑↑(↑κ a) (f ⁻¹' A) * ↑↑(↑κ a) (g ⁻¹' B) ⊢ ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) have h1 : f ⁻¹' A =ᵐ[κ a] f' ⁻¹' A := hf'.fun_comp A case h α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B a : α hf' : f =ᵐ[↑κ a] f' hg' : g =ᵐ[↑κ a] g' hfg' : ↑↑(↑κ a) (f ⁻¹' A ∩ g ⁻¹' B) = ↑↑(↑κ a) (f ⁻¹' A) * ↑↑(↑κ a) (g ⁻¹' B) h1 : f ⁻¹' A =ᵐ[↑κ a] f' ⁻¹' A ⊢ ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) have h2 : g ⁻¹' B =ᵐ[κ a] g' ⁻¹' B := hg'.fun_comp B case h α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B a : α hf' : f =ᵐ[↑κ a] f' hg' : g =ᵐ[↑κ a] g' hfg' : ↑↑(↑κ a) (f ⁻¹' A ∩ g ⁻¹' B) = ↑↑(↑κ a) (f ⁻¹' A) * ↑↑(↑κ a) (g ⁻¹' B) h1 : f ⁻¹' A =ᵐ[↑κ a] f' ⁻¹' A h2 : g ⁻¹' B =ᵐ[↑κ a] g' ⁻¹' B ⊢ ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) rwa [← measure_congr h1, ← measure_congr h2, ← measure_congr (h1.inter h2)] Qed
ProbabilityTheory.kernel.IndepFun.comp α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ ⊢ IndepFun (φ ∘ f) (ψ ∘ g) κ rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ A : Set γ hA : MeasurableSet A B : Set γ' hB : MeasurableSet B ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (φ ∘ f ⁻¹' A ∩ ψ ∘ g ⁻¹' B) = ↑↑(↑κ a) (φ ∘ f ⁻¹' A) * ↑↑(↑κ a) (ψ ∘ g ⁻¹' B) apply hfg case intro.intro.intro.intro.a α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ A : Set γ hA : MeasurableSet A B : Set γ' hB : MeasurableSet B ⊢ φ ∘ f ⁻¹' A ∈ {s \| MeasurableSet s} exact ⟨φ ⁻¹' A, hφ hA, Set.preimage_comp.symm⟩ case intro.intro.intro.intro.a α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ A : Set γ hA : MeasurableSet A B : Set γ' hB : MeasurableSet B ⊢ ψ ∘ g ⁻¹' B ∈ {s \| MeasurableSet s} exact ⟨ψ ⁻¹' B, hψ hB, Set.preimage_comp.symm⟩ Qed
ProbabilityTheory.kernel.iIndepFun.indepFun_finset α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πSβ := Set.pi (Set.univ : Set S) '' Set.pi (Set.univ : Set S) fun i => { s : Set (β i) \| MeasurableSet[m i] s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πS := { s : Set Ω \| ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπS_pi : IsPiSystem πS := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπS_gen : (MeasurableSpace.pi.comap fun a (i : S) => f i a) = generateFrom πS := by rw [generateFrom_pi.symm, comap_generateFrom] congr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πTβ := Set.pi (Set.univ : Set T) '' Set.pi (Set.univ : Set T) fun i => { s : Set (β i) \| MeasurableSet[m i] s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πT := { s : Set Ω \| ∃ t ∈ πTβ, (fun a (i : T) => f i a) ⁻¹' t = s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπT_pi : IsPiSystem πT := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπT_gen : (MeasurableSpace.pi.comap fun a (i : T) => f i a) = generateFrom πT := by rw [generateFrom_pi.symm, comap_generateFrom] congr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ refine IndepSets.indep (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) hπS_pi hπT_pi hπS_gen hπT_gen ?_ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT ⊢ IndepSets πS πT κ rintro _ _ ⟨s, ⟨sets_s, hs1, hs2⟩, rfl⟩ ⟨t, ⟨sets_t, ht1, ht2⟩, rfl⟩ ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs1 : sets_s ∈ Set.pi Set.univ fun i => {s \| MeasurableSet s} hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht1 : sets_t ∈ Set.pi Set.univ fun i => {s \| MeasurableSet s} ht2 : Set.pi Set.univ sets_t = t ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s ∩ (fun a i => f (↑i) a) ⁻¹' t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' t) simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1 case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s ∩ (fun a i => f (↑i) a) ⁻¹' t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' t) rw [← hs2, ← ht2] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IsPiSystem πS exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS ⊢ MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS rw [generateFrom_pi.symm, comap_generateFrom] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS ⊢ generateFrom ((Set.preimage fun a i => f (↑i) a) '' (Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s})) = generateFrom πS congr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IsPiSystem πT exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT ⊢ MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT rw [generateFrom_pi.symm, comap_generateFrom] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT ⊢ generateFrom ((Set.preimage fun a i => f (↑i) a) '' (Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s})) = generateFrom πT congr case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) let sets_s' : ∀ i : ι, Set (β i) := fun i => dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by intro i hi; simp_rw [dif_pos hi] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_sets_s'_univ : ∀ {i} (_hi : i ∈ T), sets_s' i = Set.univ := by intro i hi; simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) let sets_t' : ∀ i : ι, Set (β i) := fun i => dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_sets_t'_univ : ∀ {i} (_hi : i ∈ S), sets_t' i = Set.univ := by intro i hi; simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by intro i hi; rw [h_sets_s'_eq hi]; exact hs1 _ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by intro i hi; simp_rw [dif_pos hi]; exact ht1 _ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) rw [iIndepFun_iff_measure_inter_preimage_eq_mul] at hf_Indep case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) filter_upwards [hf_Indep S h_meas_s', hf_Indep T h_meas_t', hf_Indep (S ∪ T) h_meas_inter] with a h_indepS h_indepT h_indepST case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) rw [h_eq_inter_S, h_eq_inter_T, h_indepS, h_indepT, h_Inter_inter, h_indepST, Finset.prod_union hST] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ (∏ x in S, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x))) * ∏ x in T, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = (∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i)) * ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) congr 1 α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ⊢ sets_s' i = sets_s { val := i, property := hi } simp_rw [dif_pos hi] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } ⊢ ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } i : ι hi : i ∈ T ⊢ sets_s' i = Set.univ simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ⊢ sets_t' i = Set.univ simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ ⊢ ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ i : ι hi : i ∈ S ⊢ MeasurableSet (sets_s' i) rw [h_sets_s'_eq hi] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ i : ι hi : i ∈ S ⊢ MeasurableSet (sets_s { val := i, property := hi }) exact hs1 _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) ⊢ ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) i : ι hi : i ∈ T ⊢ MeasurableSet (sets_t' i) simp_rw [dif_pos hi] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) i : ι hi : i ∈ T ⊢ MeasurableSet (sets_t { val := i, property := hi }) exact ht1 _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) ⊢ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i ext1 x case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω ⊢ x ∈ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s ↔ x ∈ ⋂ i ∈ S, f i ⁻¹' sets_s' i simp_rw [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω ⊢ (∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i) ↔ ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ constructor <;> intro h case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i ⊢ ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ intro i hi case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i i : ι hi : i ∈ S ⊢ x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ simp only [h_sets_s'_eq hi, Set.mem_preimage] case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i i : ι hi : i ∈ S ⊢ f i x ∈ sets_s { val := i, property := hi } exact h ⟨i, hi⟩ case h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i rintro ⟨i, hi⟩ case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ⊢ f (↑{ val := i, property := hi }) x ∈ sets_s { val := i, property := hi } specialize h i hi case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω i : ι hi : i ∈ S h : x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f (↑{ val := i, property := hi }) x ∈ sets_s { val := i, property := hi } rwa [dif_pos hi] at h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i ⊢ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i ext1 x case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω ⊢ x ∈ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t ↔ x ∈ ⋂ i ∈ T, f i ⁻¹' sets_t' i simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω ⊢ (∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i) ↔ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ constructor <;> intro h case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i ⊢ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ intro i hi case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i i : ι hi : i ∈ T ⊢ f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ simp_rw [dif_pos hi] case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i i : ι hi : i ∈ T ⊢ f i x ∈ sets_t { val := i, property := hi } exact h ⟨i, hi⟩ case h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i rintro ⟨i, hi⟩ case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hi : i ∈ T ⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi } specialize h i hi case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω i : ι hi : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi } simp_rw [dif_pos hi] at h case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω i : ι hi : i ∈ T h : f i x ∈ sets_t { val := i, property := hi } ⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi } exact h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i ⊢ (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) ext1 x case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω ⊢ x ∈ (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i ↔ x ∈ ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) simp_rw [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω ⊢ ((∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ) ↔ ∀ (i : ι), i ∈ S ∨ i ∈ T → f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ constructor <;> intro h case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ (i : ι), i ∈ S ∨ i ∈ T → f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ intro i hi case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ∨ i ∈ T ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ cases' hi with hiS hiT case h.mp.inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hiS : i ∈ S ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ replace h := h.1 i hiS case h.mp.inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiS : i ∈ S h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ simp_rw [dif_pos hiS, dif_neg (Finset.disjoint_left.mp hST hiS)] case h.mp.inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiS : i ∈ S h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f i x ∈ sets_s { val := i, property := hiS } ∩ Set.univ exact ⟨by rwa [dif_pos hiS] at h, Set.mem_univ _⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiS : i ∈ S h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f i x ∈ sets_s { val := i, property := hiS } rwa [dif_pos hiS] at h case h.mp.inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hiT : i ∈ T ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ replace h := h.2 i hiT case h.mp.inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiT : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ simp_rw [dif_pos hiT, dif_neg (Finset.disjoint_right.mp hST hiT)] case h.mp.inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiT : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f i x ∈ Set.univ ∩ sets_t { val := i, property := hiT } exact ⟨Set.mem_univ _, by rwa [dif_pos hiT] at h⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiT : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f i x ∈ sets_t { val := i, property := hiT } rwa [dif_pos hiT] at h case h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : ∀ (i : ι), i ∈ S ∨ i ∈ T → f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) ⊢ ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) intros i hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ∪ T ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) rw [Finset.mem_union] at hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ∨ i ∈ T ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) cases' hi_mem with hi_mem hi_mem case inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) rw [h_sets_t'_univ hi_mem, Set.inter_univ] case inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ⊢ MeasurableSet (sets_s' i) exact h_meas_s' i hi_mem case inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ T ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) rw [h_sets_s'_univ hi_mem, Set.univ_inter] case inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ T ⊢ MeasurableSet (sets_t' i) exact h_meas_t' i hi_mem case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ ∏ x in S, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) refine' Finset.prod_congr rfl fun i hi => _ case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) i : ι hi : i ∈ S ⊢ ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ↑↑(↑κ a) (f i ⁻¹' sets_s' i) rw [h_sets_t'_univ hi, Set.inter_univ] case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ ∏ x in T, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) refine' Finset.prod_congr rfl fun i hi => _ case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) i : ι hi : i ∈ T ⊢ ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ↑↑(↑κ a) (f i ⁻¹' sets_t' i) rw [h_sets_s'_univ hi, Set.univ_inter] Qed
ProbabilityTheory.kernel.iIndepFun.indepFun_prod α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ classical have h_right : f k = (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘ fun a (j : ({k} : Finset ι)) => f j a := rfl have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _ let s : Finset ι := {i, j} have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘ fun a (j : s) => f j a := by ext1 a simp only [Prod.mk.inj_iff] constructor have h_meas_left : Measurable fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) := Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _) rw [h_left, h_right] refine' (hf_Indep.indepFun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right rw [Finset.disjoint_singleton_right] simp only [Finset.mem_insert, Finset.mem_singleton, not_or] exact ⟨hik.symm, hjk.symm⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_right : f k = (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘ fun a (j : ({k} : Finset ι)) => f j a := rfl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ let s : Finset ι := {i, j} α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘ fun a (j : s) => f j a := by ext1 a simp only [Prod.mk.inj_iff] constructor α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_meas_left : Measurable fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) := Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _) α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ rw [h_left, h_right] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ IndepFun ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a) ((fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a) κ refine' (hf_Indep.indepFun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ Disjoint s {k} rw [Finset.disjoint_singleton_right] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ ¬k ∈ s simp only [Finset.mem_insert, Finset.mem_singleton, not_or] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ ¬k = i ∧ ¬k = j exact ⟨hik.symm, hjk.symm⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} ⊢ (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a ext1 a case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} a : Ω ⊢ (f i a, f j a) = ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a) a simp only [Prod.mk.inj_iff] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} a : Ω ⊢ (f i a, f j a) = ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j_1 => f (↑j_1) a) a constructor ** Qed
ProbabilityTheory.kernel.iIndepFun.mul ** α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : Mul β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k ⊢ IndepFun (f i * f j) (f k) κ have : IndepFun (fun ω => (f i ω, f j ω)) (f k) κ μ := hf_Indep.indepFun_prod hf_meas i j k hik hjk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : Mul β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k this : IndepFun (fun ω => (f i ω, f j ω)) (f k) κ ⊢ IndepFun (f i * f j) (f k) κ ** change IndepFun ((fun p : β × β => p.fst * p.snd) ∘ fun ω => (f i ω, f j ω)) (id ∘ f k) κ μ ** α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : Mul β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k this : IndepFun (fun ω => (f i ω, f j ω)) (f k) κ ⊢ IndepFun ((fun p => p.1 * p.2) ∘ fun ω => (f i ω, f j ω)) (id ∘ f k) κ exact IndepFun.comp this (measurable_fst.mul measurable_snd) measurable_id Qed
ProbabilityTheory.kernel.iIndepFun.indepFun_finset_prod_of_not_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s ⊢ IndepFun (∏ j in s, f j) (f i) κ classical have h_right : f i = (fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘ fun a (j : ({i} : Finset ι)) => f j a := rfl have h_meas_right : Measurable fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ := measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩ have h_left : ∏ j in s, f j = (fun p : ∀ _j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a := by ext1 a simp only [Function.comp_apply] have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply] rw [this, Finset.prod_coe_sort] have h_meas_left : Measurable fun p : ∀ _j : s, β => ∏ j, p j := Finset.univ.measurable_prod fun (j : ↥s) (_H : j ∈ Finset.univ) => measurable_pi_apply j rw [h_left, h_right] exact (hf_Indep.indepFun_finset s {i} (Finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp h_meas_left h_meas_right α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_right : f i = (fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘ fun a (j : ({i} : Finset ι)) => f j a := rfl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_meas_right : Measurable fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ := measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_left : ∏ j in s, f j = (fun p : ∀ _j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a := by ext1 a simp only [Function.comp_apply] have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply] rw [this, Finset.prod_coe_sort] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } h_left : ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_meas_left : Measurable fun p : ∀ _j : s, β => ∏ j, p j := Finset.univ.measurable_prod fun (j : ↥s) (_H : j ∈ Finset.univ) => measurable_pi_apply j α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } h_left : ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => ∏ j : { x // x ∈ s }, p j ⊢ IndepFun (∏ j in s, f j) (f i) κ rw [h_left, h_right] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } h_left : ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => ∏ j : { x // x ∈ s }, p j ⊢ IndepFun ((fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a) ((fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a) κ exact (hf_Indep.indepFun_finset s {i} (Finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp h_meas_left h_meas_right α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } ⊢ ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a ext1 a case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω ⊢ Finset.prod s (fun j => f j) a = ((fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a) a simp only [Function.comp_apply] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω ⊢ Finset.prod s (fun j => f j) a = ∏ j : { x // x ∈ s }, f (↑j) a have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω this : ∏ j : { x // x ∈ s }, f (↑j) a = Finset.prod Finset.univ (fun j => f ↑j) a ⊢ Finset.prod s (fun j => f j) a = ∏ j : { x // x ∈ s }, f (↑j) a rw [this, Finset.prod_coe_sort] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω ⊢ ∏ j : { x // x ∈ s }, f (↑j) a = Finset.prod Finset.univ (fun j => f ↑j) a rw [Finset.prod_apply] ** Qed
ProbabilityTheory.kernel.iIndepSet.iIndepFun_indicator α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ ⊢ iIndepFun (fun _n => m) (fun n => Set.indicator (s n) fun _ω => 1) κ rw [iIndepFun_iff_measure_inter_preimage_eq_mul] α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ ⊢ ∀ (S : Finset ι) {sets : ι → Set β}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, (Set.indicator (s i) fun _ω => 1) ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) ((Set.indicator (s i) fun _ω => 1) ⁻¹' sets i) rintro S π _hπ α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, (Set.indicator (s i) fun _ω => 1) ⁻¹' π i) = ∏ i in S, ↑↑(↑κ a) ((Set.indicator (s i) fun _ω => 1) ⁻¹' π i) simp_rw [Set.indicator_const_preimage_eq_union] α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, (if 1 ∈ π i then s i else ∅) ∪ if 0 ∈ π i then (s i)ᶜ else ∅) = ∏ x in S, ↑↑(↑κ a) ((if 1 ∈ π x then s x else ∅) ∪ if 0 ∈ π x then (s x)ᶜ else ∅) refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s i)ᶜ ∅) fun i _hi => _ α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S ⊢ (fun i => (if 1 ∈ π i then s i else ∅) ∪ if 0 ∈ π i then (s i)ᶜ else ∅) i ∈ (fun x => {s_1 \| MeasurableSet s_1}) i have hsi : MeasurableSet[generateFrom {s i}] (s i) := measurableSet_generateFrom (Set.mem_singleton _) α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S hsi : MeasurableSet (s i) ⊢ (fun i => (if 1 ∈ π i then s i else ∅) ∪ if 0 ∈ π i then (s i)ᶜ else ∅) i ∈ (fun x => {s_1 \| MeasurableSet s_1}) i refine' MeasurableSet.union (MeasurableSet.ite' (fun _ => hsi) fun _ => _) (MeasurableSet.ite' (fun _ => hsi.compl) fun _ => _) case refine'_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S hsi : MeasurableSet (s i) x✝ : ¬1 ∈ π i ⊢ MeasurableSet ∅ exact @MeasurableSet.empty _ (generateFrom {s i}) case refine'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S hsi : MeasurableSet (s i) x✝ : ¬0 ∈ π i ⊢ MeasurableSet ∅ exact @MeasurableSet.empty _ (generateFrom {s i}) ** Qed
IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ⊢ Tendsto (fun j => closedBall (w j) (δ j)) l (VitaliFamily.filterAt (vitaliFamily μ K) x) refine' (vitaliFamily μ K).tendsto_filterAt_iff.mpr ⟨_, fun ε hε => _⟩ case refine'_1 α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ∈ VitaliFamily.setsAt (vitaliFamily μ K) x filter_upwards [xmem, δlim self_mem_nhdsWithin] with j hj h'j case h α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) j : ι hj : x ∈ closedBall (w j) (K * δ j) h'j : j ∈ δ ⁻¹' Ioi 0 ⊢ closedBall (w j) (δ j) ∈ VitaliFamily.setsAt (vitaliFamily μ K) x exact closedBall_mem_vitaliFamily_of_dist_le_mul μ hj h'j case refine'_2 α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε rcases l.eq_or_neBot with rfl \| h case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε ** have hK : 0 ≤ K := by rcases (xmem.and (δlim self_mem_nhdsWithin)).exists with ⟨j, hj, h'j⟩ have : 0 ≤ K * δ j := nonempty_closedBall.1 ⟨x, hj⟩ exact (mul_nonneg_iff_left_nonneg_of_pos (mem_Ioi.1 h'j)).1 this ** case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε have δpos := eventually_mem_of_tendsto_nhdsWithin δlim case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε replace δlim := tendsto_nhds_of_tendsto_nhdsWithin δlim case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε replace hK : 0 < K + 1 case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε apply (((Metric.tendsto_nhds.mp δlim _ (div_pos hε hK)).and δpos).and xmem).mono case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 ⊢ ∀ (x_1 : ι), (dist (δ x_1) 0 < ε / (K + 1) ∧ δ x_1 ∈ Ioi 0) ∧ x ∈ closedBall (w x_1) (K * δ x_1) → closedBall (w x_1) (δ x_1) ⊆ closedBall x ε rintro j ⟨⟨hjε, hj₀ : 0 < δ j⟩, hx⟩ y hy case refine'_2.inr.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hx : x ∈ closedBall (w j) (K * δ j) hjε : dist (δ j) 0 < ε / (K + 1) hj₀ : 0 < δ j y : α hy : y ∈ closedBall (w j) (δ j) ⊢ y ∈ closedBall x ε ** replace hjε : (K + 1) * δ j < ε := by simpa [abs_eq_self.mpr hj₀.le] using (lt_div_iff' hK).mp hjε ** case refine'_2.inr.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hx : x ∈ closedBall (w j) (K * δ j) hj₀ : 0 < δ j y : α hy : y ∈ closedBall (w j) (δ j) hjε : (K + 1) * δ j < ε ⊢ y ∈ closedBall x ε simp only [mem_closedBall] at hx hy ⊢ case refine'_2.inr.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hj₀ : 0 < δ j y : α hjε : (K + 1) * δ j < ε hx : dist x (w j) ≤ K * δ j hy : dist y (w j) ≤ δ j ⊢ dist y x ≤ ε linarith [dist_triangle_right y x (w j)] case refine'_2.inl α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 w : ι → α δ : ι → ℝ ε : ℝ hε : ε > 0 δlim : Tendsto δ ⊥ (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in ⊥, x ∈ closedBall (w j) (K * δ j) ⊢ ∀ᶠ (i : ι) in ⊥, closedBall (w i) (δ i) ⊆ closedBall x ε simp α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l ⊢ 0 ≤ K rcases (xmem.and (δlim self_mem_nhdsWithin)).exists with ⟨j, hj, h'j⟩ case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l j : ι hj : x ∈ closedBall (w j) (K * δ j) h'j : δ j ∈ Ioi 0 ⊢ 0 ≤ K ** have : 0 ≤ K * δ j := nonempty_closedBall.1 ⟨x, hj⟩ ** case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l j : ι hj : x ∈ closedBall (w j) (K * δ j) h'j : δ j ∈ Ioi 0 this : 0 ≤ K * δ j ⊢ 0 ≤ K exact (mul_nonneg_iff_left_nonneg_of_pos (mem_Ioi.1 h'j)).1 this case hK α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) ⊢ 0 < K + 1 linarith α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hx : x ∈ closedBall (w j) (K * δ j) hjε : dist (δ j) 0 < ε / (K + 1) hj₀ : 0 < δ j y : α hy : y ∈ closedBall (w j) (δ j) ⊢ (K + 1) * δ j < ε simpa [abs_eq_self.mpr hj₀.le] using (lt_div_iff' hK).mp hjε Qed
IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div ** α : Type u_1 inst✝⁶ : MetricSpace α inst✝⁵ : MeasurableSpace α μ : Measure α inst✝⁴ : IsUnifLocDoublingMeasure μ inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ E : Type u_2 inst✝ : NormedAddCommGroup E S : Set α K : ℝ ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ S, ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[Ioi 0] 0) → (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) → Tendsto (fun j => ↑↑μ (S ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1) filter_upwards [(vitaliFamily μ K).ae_tendsto_measure_inter_div S] with x hx ι l w δ δlim xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem) Qed
IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub ** α : Type u_1 inst✝⁶ : MetricSpace α inst✝⁵ : MeasurableSpace α μ : Measure α inst✝⁴ : IsUnifLocDoublingMeasure μ inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ E : Type u_2 inst✝ : NormedAddCommGroup E f : α → E hf : LocallyIntegrable f K : ℝ ⊢ ∀ᵐ (x : α) ∂μ, ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[Ioi 0] 0) → (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) → Tendsto (fun j => ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0) filter_upwards [(vitaliFamily μ K).ae_tendsto_average_norm_sub hf] with x hx ι l w δ δlim xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem) Qed
MeasureTheory.measurable_measure_mul_right ** G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ Measurable fun x => ↑↑μ ((fun y => y * x) ⁻¹' s) ** suffices Measurable fun y => μ ((fun x => (x, y)) ⁻¹' ((fun z : G × G => ((1 : G), z.1 * z.2)) ⁻¹' univ ×ˢ s)) by convert this using 1; ext1 x; congr 1 with y : 1; simp ** G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) apply measurable_measure_prod_mk_right case hs G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ MeasurableSet ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s) apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs) G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ MeasurableSpace G infer_instance G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) ⊢ Measurable fun x => ↑↑μ ((fun y => y * x) ⁻¹' s) convert this using 1 case h.e'_5 G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) ⊢ (fun x => ↑↑μ ((fun y => y * x) ⁻¹' s)) = fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) ext1 x case h.e'_5.h G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) x : G ⊢ ↑↑μ ((fun y => y * x) ⁻¹' s) = ↑↑μ ((fun x_1 => (x_1, x)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) congr 1 with y : 1 case h.e'_5.h.e_a.h G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) x y : G ⊢ y ∈ (fun y => y * x) ⁻¹' s ↔ y ∈ (fun x_1 => (x_1, x)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s) simp Qed
MeasureTheory.measure_inv_null G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ ⊢ ↑↑μ s⁻¹ = 0 ↔ ↑↑μ s = 0 refine' ⟨fun hs => _, (quasiMeasurePreserving_inv μ).preimage_null⟩ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ hs : ↑↑μ s⁻¹ = 0 ⊢ ↑↑μ s = 0 rw [← inv_inv s] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ hs : ↑↑μ s⁻¹ = 0 ⊢ ↑↑μ s⁻¹⁻¹ = 0 exact (quasiMeasurePreserving_inv μ).preimage_null hs ** Qed
MeasureTheory.absolutelyContinuous_inv G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ ⊢ μ ≪ Measure.inv μ refine' AbsolutelyContinuous.mk fun s _ => _ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ s : Set G x✝ : MeasurableSet s ⊢ ↑↑(Measure.inv μ) s = 0 → ↑↑μ s = 0 simp_rw [inv_apply μ s, measure_inv_null, imp_self] ** Qed
MeasureTheory.lintegral_lintegral_mul_inv ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) ⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ ** have h : Measurable fun z : G × G => (z.2 * z.1, z.1⁻¹) := (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) ⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ ** have h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν) := hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv μ ν).quasiMeasurePreserving ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ simp_rw [lintegral_lintegral h2f, lintegral_lintegral hf] G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂Measure.prod μ ν = ∫⁻ (z : G × G), f z.1 z.2 ∂Measure.prod μ ν conv_rhs => rw [← (measurePreserving_mul_prod_inv μ ν).map_eq] G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂Measure.prod μ ν = ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (Measure.prod μ ν) symm G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (Measure.prod μ ν) = ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂Measure.prod μ ν exact lintegral_map' (hf.mono' (measurePreserving_mul_prod_inv μ ν).map_eq.absolutelyContinuous) h.aemeasurable Qed
MeasureTheory.measure_mul_right_null ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ y : G ⊢ ↑↑μ ((fun x => x * y) ⁻¹' s) = 0 ↔ ↑↑μ ((fun x => y⁻¹ * x) ⁻¹' s⁻¹)⁻¹ = 0 simp_rw [← inv_preimage, preimage_preimage, mul_inv_rev, inv_inv] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ y : G ⊢ ↑↑μ ((fun x => y⁻¹ * x) ⁻¹' s⁻¹)⁻¹ = 0 ↔ ↑↑μ s = 0 simp only [measure_inv_null μ, measure_preimage_mul] Qed
MeasureTheory.absolutelyContinuous_map_mul_right ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun x => x * g) μ refine' AbsolutelyContinuous.mk fun s hs => _ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑(map (fun x => x * g) μ) s = 0 → ↑↑μ s = 0 rw [map_apply (measurable_mul_const g) hs, measure_mul_right_null] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑μ s = 0 → ↑↑μ s = 0 exact id Qed
MeasureTheory.absolutelyContinuous_map_div_left G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun h => g / h) μ simp_rw [div_eq_mul_inv] ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun h => g * h⁻¹) μ erw [← map_map (measurable_const_mul g) measurable_inv] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun x => g * x) (map Inv.inv μ) conv_lhs => rw [← map_mul_left_eq_self μ g] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ map (fun x => g * x) μ ≪ map (fun x => g * x) (map Inv.inv μ) exact (absolutelyContinuous_inv μ).map (measurable_const_mul g) Qed
MeasureTheory.absolutelyContinuous_of_isMulLeftInvariant G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 ⊢ μ ≪ ν refine' AbsolutelyContinuous.mk fun s sm hνs => _ G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 s : Set G sm : MeasurableSet s hνs : ↑↑ν s = 0 ⊢ ↑↑μ s = 0 have h1 := measure_mul_lintegral_eq μ ν sm 1 measurable_one ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 s : Set G sm : MeasurableSet s hνs : ↑↑ν s = 0 h1 : ↑↑μ s * ∫⁻ (y : G), OfNat.ofNat 1 y ∂ν = ∫⁻ (x : G), ↑↑ν ((fun z => z * x) ⁻¹' s) * OfNat.ofNat 1 x⁻¹ ∂μ ⊢ ↑↑μ s = 0 simp_rw [Pi.one_apply, lintegral_one, mul_one, (measure_mul_right_null ν _).mpr hνs, lintegral_zero, mul_eq_zero (M₀ := ℝ≥0∞), measure_univ_eq_zero.not.mpr hν, or_false_iff] at h1 G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 s : Set G sm : MeasurableSet s hνs : ↑↑ν s = 0 h1 : ↑↑μ s = 0 ⊢ ↑↑μ s = 0 exact h1 Qed
MeasureTheory.measure_mul_measure_eq ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t have h1 := measure_lintegral_div_measure ν ν hs h2s h3s (t.indicator fun _ => 1) (measurable_const.indicator ht) G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ h1 : ↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t have h2 := measure_lintegral_div_measure μ ν hs h2s h3s (t.indicator fun _ => 1) (measurable_const.indicator ht) G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ h1 : ↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν h2 : ↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ∫⁻ (x : G), indicator t (fun x => 1) x ∂μ ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t rw [lintegral_indicator _ ht, set_lintegral_one] at h1 h2 G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ h1 : ↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ↑↑ν t h2 : ↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ↑↑μ t ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t rw [← h1, mul_left_comm, h2] Qed
MeasureTheory.measurePreserving_mul_prod G : Type u_1 inst✝⁵ : MeasurableSpace G inst✝⁴ : Group G inst✝³ : MeasurableMul₂ G μ ν : Measure G inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ s : Set G inst✝ : IsMulRightInvariant μ ⊢ MeasurePreserving ?m.111270 apply measurePreserving_prod_mul_swap_right μ ν ** Qed
MeasureTheory.measurePreserving_div_prod G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulRightInvariant μ ⊢ MeasurePreserving ?m.118172 apply measurePreserving_prod_div_swap μ ν ** Qed
MeasureTheory.measurePreserving_mul_prod_inv_right ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulRightInvariant μ inst✝ : IsMulRightInvariant ν ⊢ MeasurePreserving fun z => (z.1 * z.2, z.1⁻¹) convert (measurePreserving_prod_div_swap ν μ).comp (measurePreserving_prod_mul_swap_right μ ν) using 1 case h.e'_5 G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulRightInvariant μ inst✝ : IsMulRightInvariant ν ⊢ (fun z => (z.1 * z.2, z.1⁻¹)) = (fun z => (z.2, z.1 / z.2)) ∘ fun z => (z.2, z.1 * z.2) ext1 ⟨x, y⟩ case h.e'_5.h.mk G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulRightInvariant μ inst✝ : IsMulRightInvariant ν x y : G ⊢ ((x, y).1 * (x, y).2, (x, y).1⁻¹) = ((fun z => (z.2, z.1 / z.2)) ∘ fun z => (z.2, z.1 * z.2)) (x, y) simp_rw [Function.comp_apply, div_mul_eq_div_div_swap, div_self', one_div] Qed
MeasureTheory.quasiMeasurePreserving_inv_of_right_invariant G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulRightInvariant μ ⊢ QuasiMeasurePreserving Inv.inv exact (quasiMeasurePreserving_inv μ.inv).mono (inv_absolutelyContinuous μ.inv) (absolutelyContinuous_inv μ.inv) ** Qed
MeasureTheory.quasiMeasurePreserving_div_left G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ QuasiMeasurePreserving fun h => g / h simp_rw [div_eq_mul_inv] ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ QuasiMeasurePreserving fun h => g * h⁻¹ exact (measurePreserving_mul_left μ g).quasiMeasurePreserving.comp (quasiMeasurePreserving_inv μ) Qed
MeasureTheory.quasiMeasurePreserving_mul_right ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ QuasiMeasurePreserving fun h => h * g refine' ⟨measurable_mul_const g, AbsolutelyContinuous.mk fun s hs => _⟩ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑μ s = 0 → ↑↑(map (fun h => h * g) μ) s = 0 rw [map_apply (measurable_mul_const g) hs, measure_mul_right_null] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑μ s = 0 → ↑↑μ s = 0 exact id Qed
MeasureTheory.SignedMeasure.findExistsOneDivLT_min α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i m : ℕ hm : m < MeasureTheory.SignedMeasure.findExistsOneDivLT s i ⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m rw [findExistsOneDivLT, dif_pos hi] at hm α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i m : ℕ hm : m < Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n) ⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m exact Nat.find_min _ hm ** Qed
MeasureTheory.SignedMeasure.someExistsOneDivLT_subset α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i by_cases hi : ¬s ≤[i] 0 case pos α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i ⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i exact let ⟨h, _⟩ := someExistsOneDivLT_spec hi h case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i rw [someExistsOneDivLT, dif_neg hi] case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ ∅ ⊆ i exact Set.empty_subset _ ** Qed
MeasureTheory.SignedMeasure.someExistsOneDivLT_measurableSet α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) by_cases hi : ¬s ≤[i] 0 case pos α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i ⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) exact let ⟨_, h, _⟩ := someExistsOneDivLT_spec hi h case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) rw [someExistsOneDivLT, dif_neg hi] case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ MeasurableSet ∅ exact MeasurableSet.empty ** Qed
MeasureTheory.SignedMeasure.restrictNonposSeq_succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n) = MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) rw [restrictNonposSeq] ** Qed
MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n) cases n with \| zero => rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂ rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩ exact lt_trans Nat.one_div_pos_of_nat h \| succ n => rw [restrictNonposSeq_succ] have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn convert measurable_of_not_restrict_le_zero _ hn using 3 exact funext fun x => by rw [Nat.lt_succ_iff] rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩ exact lt_trans Nat.one_div_pos_of_nat h case zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero) rw [restrictNonposSeq] case zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) rw [← @Set.diff_empty _ i] at hi₂ case zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅) hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩ case zero.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅) hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) left✝¹ : MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅) ⊆ i \ ∅ left✝ : MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) h : 1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ∅)) + 1) < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) exact lt_trans Nat.one_div_pos_of_nat h case succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n)) rw [restrictNonposSeq_succ] case succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn convert measurable_of_not_restrict_le_zero _ hn using 3 exact funext fun x => by rw [Nat.lt_succ_iff] case succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) h₁ : ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩ case succ.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) h₁ : ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) left✝¹ : MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊆ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k left✝ : MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) h : 1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) + 1) < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) exact lt_trans Nat.one_div_pos_of_nat h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ MeasurableSet (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) convert measurable_of_not_restrict_le_zero _ hn using 3 case h.e'_3.h.e'_4.h.e'_3 α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ (fun k => ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) = fun k => ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k exact funext fun x => by rw [Nat.lt_succ_iff] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k simp [Nat.lt_succ_iff] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k rfl α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) x : ℕ ⊢ ⋃ (_ : x ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x = ⋃ (_ : x < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x rw [Nat.lt_succ_iff] ** Qed
MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n < m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rw [Set.eq_empty_iff_forall_not_mem] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n < m ⊢ ∀ (x : α), ¬x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m rintro x ⟨hx₁, hx₂⟩ case intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n < m x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i m ⊢ False cases m case intro.zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n h : n < Nat.zero hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero ⊢ False rw [Nat.zero_eq] at h case intro.zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n h : n < 0 hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero ⊢ False linarith case intro.succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n n✝ : ℕ h : n < Nat.succ n✝ hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝) ⊢ False rw [restrictNonposSeq] at hx₂ case intro.succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n n✝ : ℕ h : n < Nat.succ n✝ hx₂ : x ∈ MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (H : k ≤ n✝), let_fun this := (_ : k < Nat.succ n✝); MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ False exact (someExistsOneDivLT_subset hx₂).2 (Set.mem_iUnion.2 ⟨n, Set.mem_iUnion.2 ⟨Nat.lt_succ_iff.mp h, hx₁⟩⟩) ** Qed
MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ Pairwise (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i) intro n m h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n ≠ m ⊢ (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i) n m rw [Function.onFun, Set.disjoint_iff_inter_eq_empty] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n ≠ m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rcases lt_or_gt_of_ne h with (h \| h) case inl α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h✝ : n ≠ m h : n < m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rw [restrictNonposSeq_disjoint' h] case inr α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h✝ : n ≠ m h : n > m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rw [Set.inter_comm, restrictNonposSeq_disjoint' h] ** Qed
MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ BddBelow (measureOfNegatives s) simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ ∃ x, ∀ (x_1 : ℝ), x_1 ∈ measureOfNegatives s → x ≤ x_1 by_contra' h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x ⊢ False have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x h' : ∀ (n : ℕ), ∃ y, y ∈ measureOfNegatives s ∧ y < -↑n ⊢ False choose f hf using h' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n ⊢ False have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by intro n rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩ exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n hf' : ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n ⊢ False choose B hmeas hr h_lt using hf' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n ⊢ False set A := ⋃ n, B n with hA α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n hfalse : ∀ (n : ℕ), ↑s A ≤ -↑n ⊢ False rcases exists_nat_gt (-s A) with ⟨n, hn⟩ case intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n hfalse : ∀ (n : ℕ), ↑s A ≤ -↑n n : ℕ hn : -↑s A < ↑n ⊢ False exact lt_irrefl _ ((neg_lt.1 hn).trans_le (hfalse n)) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n ⊢ ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n intro n α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n n : ℕ ⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩ case intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n n : ℕ hlt : f n < -↑n B : Set α hB₂ : ↑s B = f n hB₁ : MeasurableSet B hBr : restrict s B ≤ restrict 0 B ⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n ⊢ ∀ (n : ℕ), ↑s A ≤ -↑n intro n α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s A ≤ -↑n refine' le_trans _ (le_of_lt (h_lt _)) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s A ≤ ↑s (B n) rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n), of_union Set.disjoint_sdiff_left _ (hmeas n)] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s ((⋃ i, B i) \ B n) + ↑s (B n) ≤ ↑s (B n) refine' add_le_of_nonpos_left _ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0 have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ this : restrict s A ≤ restrict 0 A ⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0 refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ this : restrict s A ≤ restrict 0 A ⊢ MeasurableSet (⋃ i, B i) exact MeasurableSet.iUnion hmeas α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ MeasurableSet ((⋃ i, B i) \ B n) exact (MeasurableSet.iUnion hmeas).diff (hmeas n) ** Qed
MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0 rw [restrict_le_restrict_iff s 0, restrict_le_restrict_iff 0 s] at hi' hj' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0 α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet j α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet i α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet jᶜ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet iᶜ constructor α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet j α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet i α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet jᶜ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet iᶜ all_goals measurability case left α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ ↑s (i ∆ j) = 0 rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, of_union, le_antisymm (hi'.2 (hi.compl.inter hj) (Set.inter_subset_left _ _)) (hj'.1 (hi.compl.inter hj) (Set.inter_subset_right _ _)), le_antisymm (hj'.2 (hj.compl.inter hi) (Set.inter_subset_left _ _)) (hi'.1 (hj.compl.inter hi) (Set.inter_subset_right _ _)), zero_apply, zero_apply, zero_add] case left.h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ Disjoint (jᶜ ∩ i) (iᶜ ∩ j) exact Set.disjoint_of_subset_left (Set.inter_subset_left _ _) (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) (disjoint_comm.1 (IsCompl.disjoint isCompl_compl))) case left.hA α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (jᶜ ∩ i) exact hj.compl.inter hi case left.hB α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (iᶜ ∩ j) exact hi.compl.inter hj case right α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ ↑s (iᶜ ∆ jᶜ) = 0 rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, compl_compl, compl_compl, of_union, le_antisymm (hi'.2 (hj.inter hi.compl) (Set.inter_subset_right _ _)) (hj'.1 (hj.inter hi.compl) (Set.inter_subset_left _ _)), le_antisymm (hj'.2 (hi.inter hj.compl) (Set.inter_subset_right _ _)) (hi'.1 (hi.inter hj.compl) (Set.inter_subset_left _ _)), zero_apply, zero_apply, zero_add] case right.h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ Disjoint (j ∩ iᶜ) (i ∩ jᶜ) exact Set.disjoint_of_subset_left (Set.inter_subset_left _ _) (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) (IsCompl.disjoint isCompl_compl)) case right.hA α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (j ∩ iᶜ) exact hj.inter hi.compl case right.hB α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (i ∩ jᶜ) exact hi.inter hj.compl α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet iᶜ measurability ** Qed
MeasureTheory.hahn_decomposition α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let d : Set α → ℝ := fun s => ((μ s).toNNReal : ℝ) - (ν s).toNNReal α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let c : Set ℝ := d '' { s \| MeasurableSet s } α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let γ : ℝ := sSup c α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hμ : ∀ s, μ s ≠ ∞ := measure_ne_top μ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hν : ∀ s, ν s ≠ ∞ := measure_ne_top ν α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have to_nnreal_μ : ∀ s, ((μ s).toNNReal : ℝ≥0∞) = μ s := fun s => ENNReal.coe_toNNReal <\| hμ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have to_nnreal_ν : ∀ s, ((ν s).toNNReal : ℝ≥0∞) = ν s := fun s => ENNReal.coe_toNNReal <\| hν _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_split : ∀ s t, MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) := by intro s t _hs ht dsimp only rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht, ENNReal.toNNReal_add (hμ _) (hμ _), ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add, NNReal.coe_add] simp only [sub_eq_add_neg, neg_add] abel α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_Union : ∀ s : ℕ → Set α, Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) := by intro s hm refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal _).comp <\| tendsto_measure_iUnion hm exact hμ _ exact hν _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_Inter : ∀ s : ℕ → Set α, (∀ n, MeasurableSet (s n)) → (∀ n m, n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) := by intro s hs hm refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal <\| _).comp <\| tendsto_measure_iInter hs hm _ exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have bdd_c : BddAbove c := by use (μ univ).toNNReal rintro r ⟨s, _hs, rfl⟩ refine' le_trans (sub_le_self _ <\| NNReal.coe_nonneg _) _ rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ] exact measure_mono (subset_univ _) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have c_nonempty : c.Nonempty := Nonempty.image _ ⟨_, MeasurableSet.empty⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_le_γ : ∀ s, MeasurableSet s → d s ≤ γ := fun s hs => le_csSup bdd_c ⟨s, hs, rfl⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have : ∀ n : ℕ, ∃ s : Set α, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s := by intro n have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n) rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩ exact ⟨s, hs, hlt⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t rcases Classical.axiom_of_choice this with ⟨e, he⟩ case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have he₁ : ∀ n, MeasurableSet (e n) := fun n => (he n).1 case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have he₂ : ∀ n, γ - (1 / 2) ^ n < d (e n) := fun n => (he n).2 case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let f : ℕ → ℕ → Set α := fun n m => (Finset.Ico n (m + 1)).inf e case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hf : ∀ n m, MeasurableSet (f n m) := by intro n m simp only [Finset.inf_eq_iInf] exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _ case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have f_subset_f : ∀ {a b c d}, a ≤ b → c ≤ d → f a d ⊆ f b c := by intro a b c d hab hcd simp_rw [Finset.inf_eq_iInf] exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <\| Nat.succ_le_succ hcd) case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have f_succ : ∀ n m, n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) := by intro n m hnm have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm) simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm] rfl ** case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let s := ⋃ m, ⋂ n, f m n case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hs : MeasurableSet s := MeasurableSet.iUnion fun n => MeasurableSet.iInter fun m => hf _ _ case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t refine' ⟨s, hs, _, _⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s ⊢ ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) intro s t _hs ht α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ d s = d (s \ t) + d (s ∩ t) dsimp only α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) = ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \ t))) + (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t)))) rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht, ENNReal.toNNReal_add (hμ _) (hμ _), ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add, NNReal.coe_add] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - (↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑ν (s \ t)))) = ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \ t))) + (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t)))) simp only [sub_eq_add_neg, neg_add] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \ t))) + (-↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t)))) = ↑(ENNReal.toNNReal (↑↑μ (s \ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t))) + (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s ∩ t)))) abel α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) ⊢ ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) intro s hm α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal _).comp <\| tendsto_measure_iUnion hm case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ ↑↑μ (⋃ n, s n) ≠ ⊤ case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ ↑↑ν (⋃ n, s n) ≠ ⊤ exact hμ _ case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ ↑↑ν (⋃ n, s n) ≠ ⊤ exact hν _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) ⊢ ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) intro s hs hm α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal <\| _).comp <\| tendsto_measure_iInter hs hm _ case refine'_1.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ↑↑μ (⋂ n, s n) ≠ ⊤ case refine'_1.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ∃ i, ↑↑μ (s i) ≠ ⊤ case refine'_2.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ↑↑ν (⋂ n, s n) ≠ ⊤ case refine'_2.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ∃ i, ↑↑ν (s i) ≠ ⊤ exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) ⊢ BddAbove c use (μ univ).toNNReal case h α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) ⊢ ↑(ENNReal.toNNReal (↑↑μ univ)) ∈ upperBounds c rintro r ⟨s, _hs, rfl⟩ case h.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) s : Set α _hs : s ∈ {s \| MeasurableSet s} ⊢ d s ≤ ↑(ENNReal.toNNReal (↑↑μ univ)) refine' le_trans (sub_le_self _ <\| NNReal.coe_nonneg _) _ case h.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) s : Set α _hs : s ∈ {s \| MeasurableSet s} ⊢ ↑(ENNReal.toNNReal (↑↑μ s)) ≤ ↑(ENNReal.toNNReal (↑↑μ univ)) rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ] case h.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) s : Set α _hs : s ∈ {s \| MeasurableSet s} ⊢ ↑↑μ s ≤ ↑↑μ univ exact measure_mono (subset_univ _) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ ⊢ ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s intro n α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ n : ℕ ⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ n : ℕ this : γ - (1 / 2) ^ n < γ ⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩ case intro.intro.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ n : ℕ this : γ - (1 / 2) ^ n < γ s : Set α hs : s ∈ {s \| MeasurableSet s} hlt : γ - (1 / 2) ^ n < d s ⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s exact ⟨s, hs, hlt⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e ⊢ ∀ (n m : ℕ), MeasurableSet (f n m) intro n m α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e n m : ℕ ⊢ MeasurableSet (f n m) simp only [Finset.inf_eq_iInf] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e n m : ℕ ⊢ MeasurableSet (⨅ a ∈ Finset.Ico n (m + 1), e a) exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) ⊢ ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c intro a b c d hab hcd α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c✝ : Set ℝ := d✝ '' {s \| MeasurableSet s} γ : ℝ := sSup c✝ hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \ t) + d✝ (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ n, s n))) bdd_c : BddAbove c✝ c_nonempty : Set.Nonempty c✝ d_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) a b c d : ℕ hab : a ≤ b hcd : c ≤ d ⊢ f a d ⊆ f b c simp_rw [Finset.inf_eq_iInf] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c✝ : Set ℝ := d✝ '' {s \| MeasurableSet s} γ : ℝ := sSup c✝ hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \ t) + d✝ (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ n, s n))) bdd_c : BddAbove c✝ c_nonempty : Set.Nonempty c✝ d_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) a b c d : ℕ hab : a ≤ b hcd : c ≤ d ⊢ ⨅ a_1 ∈ Finset.Ico a (d + 1), e a_1 ⊆ ⨅ a ∈ Finset.Ico b (c + 1), e a exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <\| Nat.succ_le_succ hcd) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c ⊢ ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) intro n m hnm α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c n m : ℕ hnm : n ≤ m ⊢ f n (m + 1) = f n m ∩ e (m + 1) have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c n m : ℕ hnm : n ≤ m this : n ≤ m + 1 ⊢ f n (m + 1) = f n m ∩ e (m + 1) simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c n m : ℕ hnm : n ≤ m this : n ≤ m + 1 ⊢ e (m + 1) ⊓ Finset.inf (Finset.Ico n (m + 1)) e = e (m + 1) ∩ Finset.inf (Finset.Ico n (m + 1)) e rfl α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) ⊢ ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) intro n m h α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) refine' Nat.le_induction _ _ n h case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m) have := he₂ m case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n this : γ - (1 / 2) ^ m < d (e m) ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m) simp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton] case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n this : γ - (1 / 2) ^ m < d (e m) ⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) - ↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ ↑(ENNReal.toNNReal (↑↑μ (e m))) - ↑(ENNReal.toNNReal (↑↑ν (e m))) linarith case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n ⊢ ∀ (n : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1)) intro n (hmn : m ≤ n) ih case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1)) ** have : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) := by calc γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by refine' add_le_add_left (add_le_add_left _ _) γ simp only [pow_add, pow_one, le_sub_iff_add_le] linarith _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <\| he₂ _) ih) _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc] _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) := by rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left] abel exact (he₁ _).union (hf _ _) exact he₁ _ _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <\| (he₁ _).union (hf _ _)) _ ** case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) this : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1)) exact (add_le_add_iff_left γ).1 this α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) ** calc γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by refine' add_le_add_left (add_le_add_left _ _) γ simp only [pow_add, pow_one, le_sub_iff_add_le] linarith _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <\| he₂ _) ih) _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc] _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) := by rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left] abel exact (he₁ _).union (hf _ _) exact he₁ _ _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <\| (he₁ _).union (hf _ _)) _ ** α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) refine' add_le_add_left (add_le_add_left _ _) γ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ (1 / 2) ^ (n + 1) ≤ (1 / 2) ^ n - (1 / 2) ^ (n + 1) simp only [pow_add, pow_one, le_sub_iff_add_le] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ (1 / 2) ^ n * (1 / 2) + (1 / 2) ^ n * (1 / 2) ≤ (1 / 2) ^ n linarith α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) simp only [sub_eq_add_neg] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + -(2 * (1 / 2) ^ m) + ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) = sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + -(1 / 2) ^ (n + 1) + (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + -(2 * (1 / 2) ^ m) + (1 / 2) ^ n) abel α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ d (e (n + 1)) + d (f m n) ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) = d (f m n \ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1)) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1) ∪ f m n) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1)) abel case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1) ∪ f m n) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1)) exact (he₁ _).union (hf _ _) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1)) exact he₁ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n ⊢ γ ≤ d s ** have hγ : Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) := by suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by simpa only [mul_zero, tsub_zero] exact tendsto_const_nhds.sub <\| tendsto_const_nhds.mul <\| tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <\| half_pos <\| zero_lt_one) (half_lt_self zero_lt_one) ** α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ⊢ γ ≤ d s have hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) := by refine' d_Union _ _ exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <\| f_subset_f hnm <\| le_rfl α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) ⊢ γ ≤ d s refine' le_of_tendsto_of_tendsto' hγ hd fun m => _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) ⊢ γ - 2 * (1 / 2) ^ m ≤ d (⋂ n, f m n) refine' ge_of_tendsto this (eventually_atTop.2 ⟨m, fun n hmn => _⟩) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) n : ℕ hmn : n ≥ m ⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n) refine' le_trans _ (le_d_f _ _ hmn) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) n : ℕ hmn : n ≥ m ⊢ γ - 2 * (1 / 2) ^ m ≤ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n exact le_add_of_le_of_nonneg le_rfl (pow_nonneg (le_of_lt <\| half_pos <\| zero_lt_one) _) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n ⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ** suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by simpa only [mul_zero, tsub_zero] ** α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n ⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) exact tendsto_const_nhds.sub <\| tendsto_const_nhds.mul <\| tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <\| half_pos <\| zero_lt_one) (half_lt_self zero_lt_one) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n this : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) ⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) simpa only [mul_zero, tsub_zero] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ⊢ Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) refine' d_Union _ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ⊢ Monotone fun m => ⋂ n, f m n exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <\| f_subset_f hnm <\| le_rfl α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ ⊢ Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) refine' d_Inter _ _ _ case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ ⊢ ∀ (n : ℕ), MeasurableSet (f m n) intro n case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m n : ℕ ⊢ MeasurableSet (f m n) exact hf _ _ case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ ⊢ ∀ (n m_1 : ℕ), n ≤ m_1 → f m m_1 ⊆ f m n intro n m hnm case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m✝ n m : ℕ hnm : n ≤ m ⊢ f m✝ m ⊆ f m✝ n exact f_subset_f le_rfl hnm case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s ⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t intro t ht hts case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ ↑↑ν t ≤ ↑↑μ t have : 0 ≤ d t := (add_le_add_iff_left γ).1 <\| calc γ + 0 ≤ d s := by rw [add_zero]; exact γ_le_d_s _ = d (s \ t) + d t := by rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts] _ ≤ γ + d t := add_le_add (d_le_γ _ (hs.diff ht)) le_rfl case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s this : 0 ≤ d t ⊢ ↑↑ν t ≤ ↑↑μ t rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe] case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s this : 0 ≤ d t ⊢ ↑(ENNReal.toNNReal (↑↑ν t)) ≤ ↑(ENNReal.toNNReal (↑↑μ t)) simpa only [le_sub_iff_add_le, zero_add] using this α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ γ + 0 ≤ d s rw [add_zero] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ γ ≤ d s exact γ_le_d_s α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ d s = d (s \ t) + d t rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts] case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s ⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t intro t ht hts case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ ↑↑μ t ≤ ↑↑ν t have : d t ≤ 0 := (add_le_add_iff_left γ).1 <\| calc γ + d t ≤ d s + d t := add_le_add γ_le_d_s le_rfl _ = d (s ∪ t) := by rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right, (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left] _ ≤ γ + 0 := by rw [add_zero]; exact d_le_γ _ (hs.union ht) case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ this : d t ≤ 0 ⊢ ↑↑μ t ≤ ↑↑ν t rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe] case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ this : d t ≤ 0 ⊢ ↑(ENNReal.toNNReal (↑↑μ t)) ≤ ↑(ENNReal.toNNReal (↑↑ν t)) simpa only [sub_le_iff_le_add, zero_add] using this α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ d s + d t = d (s ∪ t) rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right, (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ d (s ∪ t) ≤ γ + 0 rw [add_zero] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ d (s ∪ t) ≤ γ exact d_le_γ _ (hs.union ht) Qed
measurable_div_const' G : Type u_1 inst✝² : DivInvMonoid G inst✝¹ : MeasurableSpace G inst✝ : MeasurableMul G g : G ⊢ Measurable fun h => h / g simp_rw [div_eq_mul_inv, measurable_mul_const] ** Qed
nullMeasurableSet_eq_fun G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g ⊢ NullMeasurableSet {x \| f x = g x} apply (measurableSet_eq_fun hf.measurable_mk hg.measurable_mk).nullMeasurableSet.congr G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g ⊢ {x \| AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} =ᵐ[μ] {x \| f x = g x} filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx case h G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g x : α hfx : f x = AEMeasurable.mk f hf x hgx : g x = AEMeasurable.mk g hg x ⊢ setOf (fun x => AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) x = setOf (fun x => f x = g x) x change (hf.mk f x = hg.mk g x) = (f x = g x) case h G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g x : α hfx : f x = AEMeasurable.mk f hf x hgx : g x = AEMeasurable.mk g hg x ⊢ (AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) = (f x = g x) simp only [hfx, hgx] ** Qed
measurableSet_eq_fun_of_countable G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g ⊢ MeasurableSet {x \| f x = g x} have : { x \| f x = g x } = ⋃ j, { x \| f x = j } ∩ { x \| g x = j } := by ext1 x simp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right'] G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} ⊢ MeasurableSet {x \| f x = g x} rw [this] G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} ⊢ MeasurableSet (⋃ j, {x \| f x = j} ∩ {x \| g x = j}) refine' MeasurableSet.iUnion fun j => MeasurableSet.inter _ _ G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g ⊢ {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} ext1 x case h G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g x : α ⊢ x ∈ {x \| f x = g x} ↔ x ∈ ⋃ j, {x \| f x = j} ∩ {x \| g x = j} simp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right'] case refine'_1 G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} j : E ⊢ MeasurableSet {x \| f x = j} exact hf (measurableSet_singleton j) case refine'_2 G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} j : E ⊢ MeasurableSet {x \| g x = j} exact hg (measurableSet_singleton j) ** Qed
ae_eq_trim_of_measurable G : Type u_1 α✝ : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m✝ : MeasurableSpace α✝ f✝ g✝ : α✝ → G μ✝ : Measure α✝ α : Type u_3 E : Type u_4 m m0 : MeasurableSpace α μ : Measure α inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E hm : m ≤ m0 f g : α → E hf : Measurable f hg : Measurable g hfg : f =ᵐ[μ] g ⊢ f =ᵐ[Measure.trim μ hm] g rwa [Filter.EventuallyEq, @ae_iff _ m, trim_measurableSet_eq hm _] G : Type u_1 α✝ : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m✝ : MeasurableSpace α✝ f✝ g✝ : α✝ → G μ✝ : Measure α✝ α : Type u_3 E : Type u_4 m m0 : MeasurableSpace α μ : Measure α inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E hm : m ≤ m0 f g : α → E hf : Measurable f hg : Measurable g hfg : f =ᵐ[μ] g ⊢ MeasurableSet {a \| ¬f a = g a} exact @MeasurableSet.compl α _ m (@measurableSet_eq_fun α m E _ _ _ _ _ _ hf hg) ** Qed
measurable_inv_iff G✝ : Type u_1 α : Type u_2 inst✝⁵ : Inv G✝ inst✝⁴ : MeasurableSpace G✝ inst✝³ : MeasurableInv G✝ m : MeasurableSpace α f✝ : α → G✝ μ : Measure α G : Type u_3 inst✝² : Group G inst✝¹ : MeasurableSpace G inst✝ : MeasurableInv G f : α → G h : Measurable fun x => (f x)⁻¹ ⊢ Measurable f simpa only [inv_inv] using h.inv ** Qed
aemeasurable_inv_iff G✝ : Type u_1 α : Type u_2 inst✝⁵ : Inv G✝ inst✝⁴ : MeasurableSpace G✝ inst✝³ : MeasurableInv G✝ m : MeasurableSpace α f✝ : α → G✝ μ : Measure α G : Type u_3 inst✝² : Group G inst✝¹ : MeasurableSpace G inst✝ : MeasurableInv G f : α → G h : AEMeasurable fun x => (f x)⁻¹ ⊢ AEMeasurable f simpa only [inv_inv] using h.inv ** Qed
measurable_inv_iff₀ G : Type u_1 α : Type u_2 inst✝⁵ : Inv G inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableInv G m : MeasurableSpace α f✝ : α → G μ : Measure α G₀ : Type u_3 inst✝² : GroupWithZero G₀ inst✝¹ : MeasurableSpace G₀ inst✝ : MeasurableInv G₀ f : α → G₀ h : Measurable fun x => (f x)⁻¹ ⊢ Measurable f simpa only [inv_inv] using h.inv ** Qed
aemeasurable_inv_iff₀ G : Type u_1 α : Type u_2 inst✝⁵ : Inv G inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableInv G m : MeasurableSpace α f✝ : α → G μ : Measure α G₀ : Type u_3 inst✝² : GroupWithZero G₀ inst✝¹ : MeasurableSpace G₀ inst✝ : MeasurableInv G₀ f : α → G₀ h : AEMeasurable fun x => (f x)⁻¹ ⊢ AEMeasurable f simpa only [inv_inv] using h.inv ** Qed
measurable_const_smul_iff M : Type u_1 β : Type u_2 α : Type u_3 inst✝⁹ : MeasurableSpace M inst✝⁸ : MeasurableSpace β inst✝⁷ : Monoid M inst✝⁶ : MulAction M β inst✝⁵ : MeasurableSMul M β inst✝⁴ : MeasurableSpace α f : α → β μ : Measure α G : Type u_4 inst✝³ : Group G inst✝² : MeasurableSpace G inst✝¹ : MulAction G β inst✝ : MeasurableSMul G β c : G h : Measurable fun x => c • f x ⊢ Measurable f simpa only [inv_smul_smul] using h.const_smul' c⁻¹ ** Qed
aemeasurable_const_smul_iff M : Type u_1 β : Type u_2 α : Type u_3 inst✝⁹ : MeasurableSpace M inst✝⁸ : MeasurableSpace β inst✝⁷ : Monoid M inst✝⁶ : MulAction M β inst✝⁵ : MeasurableSMul M β inst✝⁴ : MeasurableSpace α f : α → β μ : Measure α G : Type u_4 inst✝³ : Group G inst✝² : MeasurableSpace G inst✝¹ : MulAction G β inst✝ : MeasurableSMul G β c : G h : AEMeasurable fun x => c • f x ⊢ AEMeasurable f simpa only [inv_smul_smul] using h.const_smul' c⁻¹ ** Qed

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MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁷ : Group G inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α c : G μ : Measure α inst✝³ : SMulInvariantMeasure G α μ inst✝² : TopologicalSpace α inst✝¹ : ContinuousConstSMul G α inst✝ : MulAction.IsMinimal G α K U : Set α hK : IsCompact K hμK : ↑↑μ K ≠ 0 hU : IsOpen U hne : Set.Nonempty U t : Finset G ht : K ⊆ ⋃ g ∈ t, g • U hμU : ↑↑μ U = 0 x✝¹ : G x✝ : x✝¹ ∈ ↑t ⊢ ↑↑μ (x✝¹ • U) = 0 rwa [measure_smul] ** Qed
MeasureTheory.isLocallyFiniteMeasure_of_smulInvariant G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁷ : Group G inst✝⁶ : MulAction G α inst✝⁵ : MeasurableSpace G inst✝⁴ : MeasurableSMul G α c : G μ : Measure α inst✝³ : SMulInvariantMeasure G α μ inst✝² : TopologicalSpace α inst✝¹ : ContinuousConstSMul G α inst✝ : MulAction.IsMinimal G α K U : Set α hU : IsOpen U hne : Set.Nonempty U hμU : ↑↑μ U ≠ ⊤ x : α g : G hg : g • x ∈ U ⊢ ↑↑μ ((fun x x_1 => x • x_1) g ⁻¹' U) ≠ ⊤ rwa [measure_preimage_smul] ** Qed
MeasureTheory.smul_ae_eq_self_of_mem_zpowers G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x y : G hs : x • s =ᶠ[ae μ] s hy : y ∈ Subgroup.zpowers x ⊢ y • s =ᶠ[ae μ] s obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp hy case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x ⊢ x ^ k • s =ᶠ[ae μ] s let e : α ≃ α := MulAction.toPermHom G α x case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x ⊢ x ^ k • s =ᶠ[ae μ] s have he : QuasiMeasurePreserving e μ μ := (measurePreserving_smul x μ).quasiMeasurePreserving case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e ⊢ x ^ k • s =ᶠ[ae μ] s have he' : QuasiMeasurePreserving e.symm μ μ := (measurePreserving_smul x⁻¹ μ).quasiMeasurePreserving case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e he' : QuasiMeasurePreserving ↑e.symm ⊢ x ^ k • s =ᶠ[ae μ] s have h := he.image_zpow_ae_eq he' k hs case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e he' : QuasiMeasurePreserving ↑e.symm h : ↑(e ^ k) '' s =ᶠ[ae μ] s ⊢ x ^ k • s =ᶠ[ae μ] s simp only [← MonoidHom.map_zpow] at h case intro G : Type u M : Type v α : Type w s : Set α m : MeasurableSpace α inst✝⁴ : Group G inst✝³ : MulAction G α inst✝² : MeasurableSpace G inst✝¹ : MeasurableSMul G α c : G μ : Measure α inst✝ : SMulInvariantMeasure G α μ x : G hs : x • s =ᶠ[ae μ] s k : ℤ hy : x ^ k ∈ Subgroup.zpowers x e : α ≃ α := ↑(MulAction.toPermHom G α) x he : QuasiMeasurePreserving ↑e he' : QuasiMeasurePreserving ↑e.symm h : (fun a => ↑(↑(MulAction.toPermHom G α) (x ^ k)) a) '' s =ᶠ[ae μ] s ⊢ x ^ k • s =ᶠ[ae μ] s simpa only [MulAction.toPermHom_apply, MulAction.toPerm_apply, image_smul] using h ** Qed
Nat.mem_rfind p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m ⊢ n ∈ rfind p let ⟨m, hm⟩ := dom_iff_mem.1 <\| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩ p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p ⊢ n ∈ rfind p rcases lt_trichotomy m n with (h \| h \| h) case inl p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p h : m < n ⊢ n ∈ rfind p injection mem_unique (h₂ h) (rfind_spec hm) case inr.inl p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p h : m = n ⊢ n ∈ rfind p rwa [← h] case inr.inr p : ℕ →. Bool n : ℕ x✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m h₁ : true ∈ p n h₂ : ∀ {m : ℕ}, m < n → false ∈ p m m : ℕ hm : m ∈ rfind p h : n < m ⊢ n ∈ rfind p injection mem_unique h₁ (rfind_min hm h) ** Qed
Nat.rfind_min' p : ℕ → Bool m : ℕ pm : p m = true this : true ∈ ↑p m n : ℕ hn : n ∈ rfind ↑p h : m < n ⊢ False injection mem_unique this (rfind_min hn h) ** Qed
Nat.rfindOpt_dom α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n ⊢ (rfindOpt f).Dom have h' : ∃ n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2 α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true ⊢ (rfindOpt f).Dom have s := Nat.find_spec h' α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true ⊢ (rfindOpt f).Dom have fd : (rfind fun n => (f n).isSome).Dom := ⟨Nat.find h', by simpa using s.symm, fun _ _ => trivial⟩ α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom ⊢ (rfindOpt f).Dom refine' ⟨fd, _⟩ α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom ⊢ ((fun b => (fun n => ↑(f n)) (Part.get (rfind fun n => ↑(Option.some (Option.isSome (f n)))) b)) fd).Dom have := rfind_spec (get_mem fd) α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true ∈ ↑(Option.some (Option.isSome (f (Part.get (rfind fun n => ↑(Option.some (Option.isSome (f n)))) fd)))) ⊢ ((fun b => (fun n => ↑(f n)) (Part.get (rfind fun n => ↑(Option.some (Option.isSome (f n)))) b)) fd).Dom simp at this ⊢ α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true = Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) ⊢ Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) = true cases' Option.isSome_iff_exists.1 this.symm with a e case intro α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true = Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) a : α e : f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)) = Option.some a ⊢ Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) = true rw [e] case intro α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true fd : (rfind fun n => ↑(Option.some (Option.isSome (f n)))).Dom this : true = Option.isSome (f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom))) a : α e : f (Part.get (rfind fun n => Part.some (Option.isSome (f n))) (_ : (rfind fun n => Part.some (Option.isSome (f n))).Dom)) = Option.some a ⊢ Option.isSome (Option.some a) = true trivial α : Type u_1 f : ℕ → Option α h : ∃ n a, a ∈ f n h' : ∃ n, Option.isSome (f n) = true s : Option.isSome (f (Nat.find h')) = true ⊢ true ∈ (fun n => ↑(Option.some (Option.isSome (f n)))) (Nat.find h') simpa using s.symm ** Qed
Nat.rfindOpt_mono α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n ⊢ a ∈ rfindOpt f have h' := rfindOpt_dom.2 ⟨_, _, h⟩ α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom ⊢ a ∈ rfindOpt f cases' rfindOpt_spec ⟨h', rfl⟩ with k hk case intro α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom k : ℕ hk : Part.get (rfindOpt f) h' ∈ f k ⊢ a ∈ rfindOpt f have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk) case intro α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom k : ℕ hk : Part.get (rfindOpt f) h' ∈ f k this : Option.some a = Option.some (Part.get (rfindOpt f) h') ⊢ a ∈ rfindOpt f simp at this case intro α : Type u_1 f : ℕ → Option α H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n a : α x✝ : ∃ n, a ∈ f n n : ℕ h : a ∈ f n h' : (rfindOpt f).Dom k : ℕ hk : Part.get (rfindOpt f) h' ∈ f k this : a = Part.get (rfindOpt f) h' ⊢ a ∈ rfindOpt f simp [this, get_mem] ** Qed
Nat.Partrec.of_primrec f : ℕ → ℕ hf : Nat.Primrec f ⊢ Partrec ↑f induction hf with \| zero => exact zero \| succ => exact succ \| left => exact left \| right => exact right \| pair _ _ pf pg => refine' (pf.pair pg).of_eq_tot fun n => _ simp [Seq.seq] \| comp _ _ pf pg => refine' (pf.comp pg).of_eq_tot fun n => _ simp \| prec _ _ pf pg => refine' (pf.prec pg).of_eq_tot fun n => _ simp only [unpaired, PFun.coe_val, bind_eq_bind] induction n.unpair.2 with \| zero => simp \| succ m IH => simp only [mem_bind_iff, mem_some_iff] exact ⟨_, IH, rfl⟩ case zero f : ℕ → ℕ ⊢ Partrec ↑fun x => 0 exact zero case succ f : ℕ → ℕ ⊢ Partrec ↑Nat.succ exact succ case left f : ℕ → ℕ ⊢ Partrec ↑fun n => (unpair n).1 exact left case right f : ℕ → ℕ ⊢ Partrec ↑fun n => (unpair n).2 exact right case pair f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ ⊢ Partrec ↑fun n => Nat.pair (f✝ n) (g✝ n) refine' (pf.pair pg).of_eq_tot fun n => _ case pair f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ Nat.pair (f✝ n) (g✝ n) ∈ Seq.seq (Nat.pair <$> ↑f✝ n) fun x => ↑g✝ n simp [Seq.seq] case comp f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ ⊢ Partrec ↑fun n => f✝ (g✝ n) refine' (pf.comp pg).of_eq_tot fun n => _ case comp f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ f✝ (g✝ n) ∈ ↑g✝ n >>= ↑f✝ simp case prec f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ ⊢ Partrec ↑(unpaired fun z n => Nat.rec (f✝ z) (fun y IH => g✝ (Nat.pair z (Nat.pair y IH))) n) refine' (pf.prec pg).of_eq_tot fun n => _ case prec f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ unpaired (fun z n => Nat.rec (f✝ z) (fun y IH => g✝ (Nat.pair z (Nat.pair y IH))) n) n ∈ unpaired (fun a n => Nat.rec (↑f✝ a) (fun y IH => do let i ← IH ↑g✝ (Nat.pair a (Nat.pair y i))) n) n simp only [unpaired, PFun.coe_val, bind_eq_bind] case prec f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) (unpair n).2 ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) (unpair n).2 induction n.unpair.2 with \| zero => simp \| succ m IH => simp only [mem_bind_iff, mem_some_iff] exact ⟨_, IH, rfl⟩ case prec.zero f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n : ℕ ⊢ Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) Nat.zero ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) Nat.zero simp case prec.succ f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n m : ℕ IH : Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) m ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) m ⊢ Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) (Nat.succ m) ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) (Nat.succ m) simp only [mem_bind_iff, mem_some_iff] case prec.succ f f✝ g✝ : ℕ → ℕ a✝¹ : Nat.Primrec f✝ a✝ : Nat.Primrec g✝ pf : Partrec ↑f✝ pg : Partrec ↑g✝ n m : ℕ IH : Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) m ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) m ⊢ ∃ a, a ∈ Nat.rec (Part.some (f✝ (unpair n).1)) (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).1 (Nat.pair y i)))) m ∧ g✝ (Nat.pair (unpair n).1 (Nat.pair m (Nat.rec (f✝ (unpair n).1) (fun y IH => g✝ (Nat.pair (unpair n).1 (Nat.pair y IH))) m))) = g✝ (Nat.pair (unpair n).1 (Nat.pair m a)) exact ⟨_, IH, rfl⟩ ** Qed
Nat.Partrec.none n a : ℕ x✝ : a ∈ Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (↑fun x => 1) (Nat.pair n n_1) h : (Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (↑fun x => 1) (Nat.pair n n_1)).Dom h✝ : Part.get (Nat.rfind fun n_1 => (fun m => decide (m = 0)) <$> (↑fun x => 1) (Nat.pair n n_1)) h = a ⊢ False simp at h ** Qed
Primrec.to_comp α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → σ hf : Primrec f n : ℕ ⊢ (do let n ← (↑fun n => encode (Option.map f (decode n))) n ↑(Nat.ppred n)) = Part.bind ↑(decode n) fun a => map encode (↑f a) simp α : Type u_1 σ : Type u_2 inst✝¹ : Primcodable α inst✝ : Primcodable σ f : α → σ hf : Primrec f n : ℕ ⊢ ↑(Nat.ppred (encode (Option.map f (decode n)))) = Part.bind ↑(decode n) fun a => Part.some (encode (f a)) cases decode (α := α) n <;> simp ** Qed
Computable.ofOption α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β hf : Computable f n : ℕ ⊢ (do let n ← Part.bind ↑(decode n) fun a => map encode (↑f a) ↑(Nat.ppred n)) = Part.bind ↑(decode n) fun a => map encode ((fun a => ↑(f a)) a) cases' decode (α := α) n with a <;> simp case some α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β hf : Computable f n : ℕ a : α ⊢ ↑(Nat.ppred (encode (f a))) = map encode ↑(f a) cases' f a with b <;> simp ** Qed
Computable.pair α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β g : α → γ hf : Computable f hg : Computable g n : ℕ ⊢ (Seq.seq (Nat.pair <$> Part.bind ↑(decode n) fun a => map encode (↑f a)) fun x => Part.bind ↑(decode n) fun a => map encode (↑g a)) = Part.bind ↑(decode n) fun a => map encode ((↑fun a => (f a, g a)) a) cases decode (α := α) n <;> simp [Seq.seq] ** Qed
Partrec.none α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ ⊢ Part.none = Part.bind ↑(decode n) fun a => map encode ((fun x => Part.none) a) cases decode (α := α) n <;> simp ** Qed
Partrec.bind α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. β g : α → β →. σ hf : Partrec f hg : Partrec₂ g n : ℕ ⊢ (do let n ← Seq.seq (Nat.pair <$> Part.some n) fun x => Part.bind ↑(decode n) fun a => map encode (f a) Part.bind ↑(decode n) fun a => map encode ((fun p => g p.1 p.2) a)) = Part.bind ↑(decode n) fun a => map encode ((fun a => Part.bind (f a) (g a)) a) simp [Seq.seq] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. β g : α → β →. σ hf : Partrec f hg : Partrec₂ g n : ℕ ⊢ (Part.bind (Part.bind ↑(decode n) fun y => map (Nat.pair n) (map encode (f y))) fun n => Part.bind ↑(Option.bind (decode (Nat.unpair n).1) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).2)) fun a => map encode (g a.1 a.2)) = Part.bind ↑(decode n) fun a => Part.bind (f a) fun y => map encode (g a y) cases' e : decode (α := α) n with a <;> simp [e, encodek] ** Qed
Partrec.map α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. β g : α → β → σ hf : Partrec f hg : Computable₂ g ⊢ Partrec fun a => Part.map (g a) (f a) simpa [bind_some_eq_map] using @Partrec.bind _ _ _ _ _ _ _ (fun a => Part.some ∘ (g a)) hf hg ** Qed
Partrec.comp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : β →. σ g : α → β hf : Partrec f hg : Computable g n : ℕ ⊢ (do let n ← Part.bind ↑(decode n) fun a => Part.map encode (↑g a) Part.bind ↑(decode n) fun a => Part.map encode (f a)) = Part.bind ↑(decode n) fun a => Part.map encode ((fun a => f (g a)) a) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : β →. σ g : α → β hf : Partrec f hg : Computable g n : ℕ ⊢ (Part.bind (Part.bind ↑(decode n) fun a => Part.some (encode (g a))) fun n => Part.bind ↑(decode n) fun a => Part.map encode (f a)) = Part.bind ↑(decode n) fun a => Part.map encode (f (g a)) cases' e : decode (α := α) n with a <;> simp [e, encodek] ** Qed
Partrec.nat_iff α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : ℕ →. ℕ ⊢ Partrec f ↔ Nat.Partrec f simp [Partrec, map_id'] ** Qed
Partrec₂.unpaired α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 σ : Type u_5 inst✝⁴ : Primcodable α inst✝³ : Primcodable β inst✝² : Primcodable γ inst✝¹ : Primcodable δ inst✝ : Primcodable σ f : ℕ → ℕ →. α h : Partrec (Nat.unpaired f) ⊢ Partrec₂ f simpa using Partrec.comp (g := fun p : ℕ × ℕ => (p.1, p.2)) h Primrec₂.pair.to_comp ** Qed
Partrec.nat_casesOn_right α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α ⊢ Nat.rec (↑g a) (fun y IH => Part.bind IH fun i => h (a, y, i).1 (Nat.pred (f (a, y, i).1))) (f a) = Nat.casesOn (f a) (Part.some (g a)) (h a) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α ⊢ Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a (Nat.pred (f a))) (f a) = Nat.rec (Part.some (g a)) (fun n n_ih => h a n) (f a) cases' f a with n <;> simp case succ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ ⊢ (Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n) = h a n refine' ext fun b => ⟨fun H => _, fun H => _⟩ case succ.refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n ⊢ b ∈ h a n rcases mem_bind_iff.1 H with ⟨c, _, h₂⟩ case succ.refine'_1.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n c : σ left✝ : c ∈ Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n h₂ : b ∈ h a n ⊢ b ∈ h a n exact h₂ case succ.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n ⊢ b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n have : ∀ m, (Nat.rec (motive := fun _ => Part σ) (Part.some (g a)) (fun y IH => IH.bind fun _ => h a n) m).Dom := by intro m induction m <;> simp [, H.fst] * case succ.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n this : ∀ (m : ℕ), (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom ⊢ b ∈ Part.bind (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => h a n) n) fun i => h a n exact ⟨⟨this n, H.fst⟩, H.snd⟩ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n ⊢ ∀ (m : ℕ), (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom intro m α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ →. σ hf : Computable f hg : Computable g hh : Partrec₂ h a : α n : ℕ b : σ H : b ∈ h a n m : ℕ ⊢ (Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun x => h a n) m).Dom ** induction m <;> simp [, H.fst] * Qed
Partrec.vector_mOfFn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α →. σ hf : ∀ (i : Fin (n + 1)), Partrec (f i) ⊢ Partrec fun a => Vector.mOfFn fun i => f i a simp only [Vector.mOfFn, Nat.add_eq, Nat.add_zero, pure_eq_some, bind_eq_bind] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α →. σ hf : ∀ (i : Fin (n + 1)), Partrec (f i) ⊢ Partrec fun a => Part.bind (f 0 a) fun a_1 => Part.bind (Vector.mOfFn fun i => f (Fin.succ i) a) fun v => Part.some (a_1 ::ᵥ v) exact (hf 0).bind (Partrec.bind ((vector_mOfFn fun i => hf i.succ).comp fst) (Primrec.vector_cons.to_comp.comp (snd.comp fst) snd)) ** Qed
Computable.bind_decode_iff α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ fun a n => Option.bind (decode n) (f a) n : ℕ ⊢ (Part.bind ↑(decode n) fun a => map encode ((fun a => Part.bind (do let n ← (↑fun n => encode (decode n)) a.2 ↑(Nat.ppred n)) ↑fun b => (fun a n => Option.bind (decode n) (f a)) (a, b).1.1 (a, b).1.2) a)) = Part.bind ↑(decode n) fun a => map encode ((↑fun p => f p.1 p.2) a) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ fun a n => Option.bind (decode n) (f a) n : ℕ ⊢ (Part.bind ↑(Option.bind (decode (Nat.unpair n).1) fun a => Option.some (a, (Nat.unpair n).2)) fun a => Part.bind ↑(Nat.ppred (encode (decode a.2))) fun y => Part.some (encode (Option.bind (decode a.2) (f a.1)))) = Part.bind ↑(Option.bind (decode (Nat.unpair n).1) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).2)) fun a => Part.some (encode (f a.1 a.2)) cases decode (α := α) n.unpair.1 <;> simp case some α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ fun a n => Option.bind (decode n) (f a) n : ℕ val✝ : α ⊢ (Part.bind ↑(Nat.ppred (encode (decode (Nat.unpair n).2))) fun y => Part.some (encode (Option.bind (decode (Nat.unpair n).2) (f val✝)))) = Part.bind ↑(Option.map (Prod.mk val✝) (decode (Nat.unpair n).2)) fun a => Part.some (encode (f a.1 a.2)) cases decode (α := β) n.unpair.2 <;> simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f ⊢ Computable₂ fun a n => Option.bind (decode n) (f a) have : Partrec fun a : α × ℕ => (encode (decode (α := β) a.2)).casesOn (some Option.none) fun n => Part.map (f a.1) (decode (α := β) n) := Partrec.nat_casesOn_right (h := fun (a : α × ℕ) (n : ℕ) ↦ map (fun b ↦ f a.1 b) (Part.ofOption (decode n))) (Primrec.encdec.to_comp.comp snd) (const Option.none) ((ofOption (Computable.decode.comp snd)).map (hf.comp (fst.comp <\| fst.comp fst) snd).to₂) α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f this : Partrec fun a => Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n) ⊢ Computable₂ fun a n => Option.bind (decode n) (f a) refine' this.of_eq fun a => _ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f this : Partrec fun a => Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n) a : α × ℕ ⊢ (Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n)) = (↑fun p => (fun a n => Option.bind (decode n) (f a)) p.1 p.2) a simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β → Option σ hf : Computable₂ f this : Partrec fun a => Nat.casesOn (encode (decode a.2)) (Part.some Option.none) fun n => map (f a.1) ↑(decode n) a : α × ℕ ⊢ Nat.rec (Part.some Option.none) (fun n n_ih => map (f a.1) ↑(decode n)) (encode (decode a.2)) = Part.some (Option.bind (decode a.2) (f a.1)) cases decode (α := β) a.2 <;> simp [encodek] ** Qed
Computable.nat_rec α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ × σ → σ hf : Computable f hg : Computable g hh : Computable₂ h a : α ⊢ Nat.rec (↑g a) (fun y IH => Part.bind IH fun i => ↑(h a) (y, i)) (f a) = (↑fun a => Nat.rec (g a) (fun y IH => h a (y, IH)) (f a)) a simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ g : α → σ h : α → ℕ × σ → σ hf : Computable f hg : Computable g hh : Computable₂ h a : α ⊢ Nat.rec (Part.some (g a)) (fun y IH => Part.bind IH fun i => Part.some (h a (y, i))) (f a) = Part.some (Nat.rec (g a) (fun y IH => h a (y, IH)) (f a)) induction f a <;> simp [] * Qed
Computable.cond α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ c : α → Bool f g : α → σ hc : Computable c hf : Computable f hg : Computable g a : α ⊢ (Nat.casesOn (encode (c a)) (g a) fun b => f (a, b).1) = bif c a then f a else g a cases c a <;> rfl ** Qed
Computable.option_bind α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β → Option σ hf : Computable f hg : Computable₂ g a : α ⊢ Option.casesOn (f a) Option.none (g a) = Option.bind (f a) (g a) cases f a <;> rfl ** Qed
Computable.option_map α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β → σ hf : Computable f hg : Computable₂ g ⊢ Computable fun a => Option.map (g a) (f a) convert option_bind hf (option_some.comp₂ hg) case h.e'_5.h α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β → σ hf : Computable f hg : Computable₂ g x✝ : α ⊢ Option.map (g x✝) (f x✝) = Option.bind (f x✝) fun b => Option.some (g x✝ b) apply Option.map_eq_bind ** Qed
Computable.option_getD α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → Option β g : α → β hf : Computable f hg : Computable g a : α ⊢ (Option.casesOn (f a) (g a) fun b => b) = Option.getD (f a) (g a) cases f a <;> rfl ** Qed
Computable.sum_casesOn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β ⊕ γ g : α → β → σ h : α → γ → σ hf : Computable f hg : Computable₂ g hh : Computable₂ h a : α ⊢ (bif Nat.bodd (encode (f a)) then Option.map (h a) (decode (Nat.div2 (encode (f a)))) else Option.map (g a) (decode (Nat.div2 (encode (f a))))) = Option.some (Sum.casesOn (f a) (g a) (h a)) cases' f a with b c <;> simp [Nat.div2_val] ** Qed
Computable.nat_strong_rec α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) this : Computable₂ fun a n => List.map (f a) (List.range n) a : α × ℕ ⊢ (fun a b => List.get? (List.map (f (a, b).1) (List.range (Nat.succ (a, b).2))) (a, b).2) a.1 a.2 = Option.some (f a.1 a.2) simp [List.get?_range (Nat.lt_succ_self a.2)] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind (a, y, IH).2.2 fun b => Option.map (fun b_1 => (((a, y, IH), b), b_1).1.2 ++ [(((a, y, IH), b), b_1).2]) (g ((a, y, IH), b).1.1.1 ((a, y, IH), b).2)) a.2 = Option.some ((fun a n => List.map (f a) (List.range n)) a.1 a.2) simp α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) a.2 = Option.some (List.map (f a.1) (List.range a.2)) induction' a.2 with n IH case succ α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ n : ℕ IH : Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) n = Option.some (List.map (f a.1) (List.range n)) ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) (Nat.succ n) = Option.some (List.map (f a.1) (List.range (Nat.succ n))) simp [IH, H, List.range_succ] case zero α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → ℕ → σ g : α → List σ → Option σ hg : Computable₂ g H : ∀ (a : α) (n : ℕ), g a (List.map (f a) (List.range n)) = Option.some (f a n) a : α × ℕ ⊢ Nat.rec (Option.some []) (fun y IH => Option.bind IH fun b => Option.map (fun b_1 => b ++ [b_1]) (g a.1 b)) Nat.zero = Option.some (List.map (f a.1) (List.range Nat.zero)) rfl ** Qed
Computable.list_ofFn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α → σ hf : ∀ (i : Fin (n + 1)), Computable (f i) ⊢ Computable fun a => List.ofFn fun i => f i a simp only [List.ofFn_succ] α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin (n + 1) → α → σ hf : ∀ (i : Fin (n + 1)), Computable (f i) ⊢ Computable fun a => f 0 a :: List.ofFn fun i => f (Fin.succ i) a exact list_cons.comp (hf 0) (list_ofFn fun i => hf i.succ) ** Qed
Computable.vector_ofFn α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ n : ℕ f : Fin n → α → σ hf : ∀ (i : Fin n), Computable (f i) a : α ⊢ (Vector.mOfFn fun i => ↑(f i) a) = (↑fun a => Vector.ofFn fun i => f i a) a simp ** Qed
Partrec.option_some_iff α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α →. σ h : Partrec fun a => Part.map Option.some (f a) n : ℕ ⊢ (do let n ← Part.bind ↑(decode n) fun a => Part.map encode ((fun a => Part.map Option.some (f a)) a) ↑(Nat.ppred n)) = Part.bind ↑(decode n) fun a => Part.map encode (f a) simp [Part.bind_assoc, ← Function.comp_apply (f := Part.some) (g := encode), bind_some_eq_map, -Function.comp_apply] ** Qed
Partrec.option_casesOn_right α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ o : α → Option β f : α → σ g : α → β →. σ ho : Computable o hf : Computable f hg : Partrec₂ g this : Partrec fun a => Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (↑(decode n)) (g a) a : α ⊢ (Nat.casesOn (encode (o a)) (Part.some (f a)) fun n => Part.bind (↑(decode n)) (g a)) = Option.casesOn (o a) (Part.some (f a)) (g a) cases' o a with b <;> simp [encodek] ** Qed
Partrec.sum_casesOn_left α : Type u_1 β : Type u_2 γ : Type u_3 σ : Type u_4 inst✝³ : Primcodable α inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ f : α → β ⊕ γ g : α → β →. σ h : α → γ → σ hf : Computable f hg : Partrec₂ g hh : Computable₂ h a : α ⊢ Sum.casesOn (Sum.casesOn (f a) (fun b => Sum.inr (a, b).2) fun b => Sum.inl (a, b).2) (fun b => Part.some (h a b)) (g a) = Sum.casesOn (f a) (g a) fun c => Part.some (h a c) cases f a <;> simp ** Qed
Partrec.fix_aux α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ ⊢ let F := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n; (∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n) ↔ b ∈ PFun.fix f a intro F α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n ⊢ (∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n) ↔ b ∈ PFun.fix f a refine' ⟨fun h => _, fun h => _⟩ case refine'_1 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n ⊢ b ∈ PFun.fix f a rcases h with ⟨n, ⟨_x, h₁⟩, h₂⟩ case refine'_1.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a ⊢ b ∈ PFun.fix f a cases n <;> simp at h₂ case refine'_1.intro.intro.intro.succ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m h₂ : Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ ∨ ∃ b_1, Sum.inr b_1 ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ ∧ Sum.inl b ∈ f b_1 ⊢ b ∈ PFun.fix f a rcases h₂ with (h₂ \| ⟨a', am', fa'⟩) α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m ⊢ ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a intro m a' am ba α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m : ℕ a' : α am : Sum.inr a' ∈ F a m ba : b ∈ PFun.fix f a' ⊢ b ∈ PFun.fix f a induction' m with m IH generalizing a' <;> simp at am case succ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m✝ : ℕ a'✝ : α am✝ : Sum.inr a'✝ ∈ F a m✝ ba✝ : b ∈ PFun.fix f a'✝ m : ℕ IH : ∀ (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a a' : α ba : b ∈ PFun.fix f a' am : ∃ b, Sum.inr b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m ∧ Sum.inr a' ∈ f b ⊢ b ∈ PFun.fix f a rcases am with ⟨a₂, am₂, fa₂⟩ case succ.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m✝ : ℕ a'✝ : α am : Sum.inr a'✝ ∈ F a m✝ ba✝ : b ∈ PFun.fix f a'✝ m : ℕ IH : ∀ (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a a' : α ba : b ∈ PFun.fix f a' a₂ : α am₂ : Sum.inr a₂ ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) m fa₂ : Sum.inr a' ∈ f a₂ ⊢ b ∈ PFun.fix f a exact IH _ am₂ (PFun.mem_fix_iff.2 (Or.inr ⟨_, fa₂, ba⟩)) case zero α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n n : ℕ h₂ : Sum.inl b ∈ F a n _x : ∃ b', Sum.inl b' ∈ F a n h₁ : ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m m : ℕ a'✝ : α am✝ : Sum.inr a'✝ ∈ F a m ba✝ : b ∈ PFun.fix f a'✝ a' : α ba : b ∈ PFun.fix f a' am : a' = a ⊢ b ∈ PFun.fix f a rwa [← am] case refine'_1.intro.intro.intro.succ.inl α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m h₂ : Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ ⊢ b ∈ PFun.fix f a cases' h₁ (Nat.lt_succ_self _) with a' h case refine'_1.intro.intro.intro.succ.inl.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m h₂ : Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ a' : α h : Sum.inr a' ∈ F a n✝ ⊢ b ∈ PFun.fix f a injection mem_unique h h₂ case refine'_1.intro.intro.intro.succ.inr.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n this : ∀ (m : ℕ) (a' : α), Sum.inr a' ∈ F a m → b ∈ PFun.fix f a' → b ∈ PFun.fix f a n✝ : ℕ _x : ∃ b', Sum.inl b' ∈ F a (Nat.succ n✝) h₁ : ∀ {m : ℕ}, m < Nat.succ n✝ → ∃ b, Sum.inr b ∈ F a m a' : α am' : Sum.inr a' ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) n✝ fa' : Sum.inl b ∈ f a' ⊢ b ∈ PFun.fix f a exact this _ _ am' (PFun.mem_fix_iff.2 (Or.inl fa')) case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a ⊢ ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n suffices ∀ (a') (_ : b ∈ PFun.fix f a') (k) (_ : Sum.inr a' ∈ F a k), ∃ n, Sum.inl b ∈ F a n ∧ ∀ m < n, ∀ (_ : k ≤ m), ∃ a₂, Sum.inr a₂ ∈ F a m by rcases this _ h 0 (by simp) with ⟨n, hn₁, hn₂⟩ exact ⟨_, ⟨⟨_, hn₁⟩, fun {m} mn => hn₂ m mn (Nat.zero_le _)⟩, hn₁⟩ case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a ⊢ ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m intro a₁ h₁ case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ ⊢ ∀ (k : ℕ), Sum.inr a₁ ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m apply @PFun.fixInduction _ _ _ _ _ _ h₁ case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ ⊢ ∀ (a' : α), b ∈ PFun.fix f a' → (∀ (a'' : α), Sum.inr a'' ∈ f a' → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m) → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m intro a₂ h₂ IH k hk case refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m rcases PFun.mem_fix_iff.1 h₂ with (h₂ \| ⟨a₃, am₃, _⟩) α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a this : ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n rcases this _ h 0 (by simp) with ⟨n, hn₁, hn₂⟩ case intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a this : ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → 0 ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ ∃ n, ((∃ b', Sum.inl b' ∈ F a n) ∧ ∀ {m : ℕ}, m < n → ∃ b, Sum.inr b ∈ F a m) ∧ Sum.inl b ∈ F a n exact ⟨_, ⟨⟨_, hn₁⟩, fun {m} mn => hn₂ m mn (Nat.zero_le _)⟩, hn₁⟩ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a this : ∀ (a' : α), b ∈ PFun.fix f a' → ∀ (k : ℕ), Sum.inr a' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ Sum.inr a ∈ F a 0 simp case refine'_2.inl α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m refine' ⟨k.succ, _, fun m mk km => ⟨a₂, _⟩⟩ case refine'_2.inl.refine'_1 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ ⊢ Sum.inl b ∈ F a (Nat.succ k) simp case refine'_2.inl.refine'_1 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ ⊢ Sum.inl b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k ∨ ∃ b_1, Sum.inr b_1 ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k ∧ Sum.inl b ∈ f b_1 exact Or.inr ⟨_, hk, h₂⟩ case refine'_2.inl.refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂✝ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k h₂ : Sum.inl b ∈ f a₂ m : ℕ mk : m < Nat.succ k km : k ≤ m ⊢ Sum.inr a₂ ∈ F a m rwa [le_antisymm (Nat.le_of_lt_succ mk) km] case refine'_2.inr.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m rcases IH _ am₃ k.succ (by simp; exact ⟨_, hk, am₃⟩) with ⟨n, hn₁, hn₂⟩ α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ ⊢ Sum.inr a₃ ∈ F a (Nat.succ k) simp α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ ⊢ ∃ b, Sum.inr b ∈ Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.rec (fun val => Part.some s) (fun val => f val) s) k ∧ Sum.inr a₃ ∈ f b exact ⟨_, hk, am₃⟩ case refine'_2.inr.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m ⊢ ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m refine' ⟨n, hn₁, fun m mn km => _⟩ case refine'_2.inr.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m m : ℕ mn : m < n km : k ≤ m ⊢ ∃ a₂, Sum.inr a₂ ∈ F a m cases' km.lt_or_eq_dec with km km case refine'_2.inr.intro.intro.intro.intro.inl α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m m : ℕ mn : m < n km✝ : k ≤ m km : k < m ⊢ ∃ a₂, Sum.inr a₂ ∈ F a m exact hn₂ _ mn km case refine'_2.inr.intro.intro.intro.intro.inr α✝ : Type u_1 β : Type u_2 γ : Type u_3 σ✝ : Type u_4 inst✝³ : Primcodable α✝ inst✝² : Primcodable β inst✝¹ : Primcodable γ inst✝ : Primcodable σ✝ α : Type u_5 σ : Type u_6 f : α →. σ ⊕ α a : α b : σ F : α → ℕ →. σ ⊕ α := fun a n => Nat.rec (Part.some (Sum.inr a)) (fun x IH => Part.bind IH fun s => Sum.casesOn s (fun x => Part.some s) f) n h : b ∈ PFun.fix f a a₁ : α h₁ : b ∈ PFun.fix f a₁ a₂ : α h₂ : b ∈ PFun.fix f a₂ IH : ∀ (a'' : α), Sum.inr a'' ∈ f a₂ → ∀ (k : ℕ), Sum.inr a'' ∈ F a k → ∃ n, Sum.inl b ∈ F a n ∧ ∀ (m : ℕ), m < n → k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m k : ℕ hk : Sum.inr a₂ ∈ F a k a₃ : α am₃ : Sum.inr a₃ ∈ f a₂ right✝ : b ∈ PFun.fix f a₃ n : ℕ hn₁ : Sum.inl b ∈ F a n hn₂ : ∀ (m : ℕ), m < n → Nat.succ k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m m : ℕ mn : m < n km✝ : k ≤ m km : k = m ⊢ ∃ a₂, Sum.inr a₂ ∈ F a m exact km ▸ ⟨_, hk⟩ ** Qed
ProbabilityTheory.kernel.IndepSets.symm α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α s₁ s₂ : Set (Set Ω) h : IndepSets s₁ s₂ κ ⊢ IndepSets s₂ s₁ κ intros t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α s₁ s₂ : Set (Set Ω) h : IndepSets s₁ s₂ κ t1 t2 : Set Ω ht1 : t1 ∈ s₂ ht2 : t2 ∈ s₁ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 filter_upwards [h t2 t1 ht2 ht1] with a ha case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α s₁ s₂ : Set (Set Ω) h : IndepSets s₁ s₂ κ t1 t2 : Set Ω ht1 : t1 ∈ s₂ ht2 : t2 ∈ s₁ a : α ha : ↑↑(↑κ a) (t2 ∩ t1) = ↑↑(↑κ a) t2 * ↑↑(↑κ a) t1 ⊢ ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rwa [Set.inter_comm, mul_comm] Qed
ProbabilityTheory.kernel.indep_bot_right α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ ⊢ Indep m' ⊥ κ intros s t _ ht ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} ht : t ∈ {s \| MeasurableSet s} ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} ht : t = ∅ ∨ t = Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t refine Filter.eventually_of_forall (fun a ↦ ?_) α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} ht : t = ∅ ∨ t = Set.univ a : α ⊢ ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t cases' ht with ht ht case inl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} a : α ht : t = ∅ ⊢ ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t rw [ht, Set.inter_empty, measure_empty, mul_zero] case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m' _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s t : Set Ω a✝ : s ∈ {s \| MeasurableSet s} a : α ht : t = Set.univ ⊢ ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t rw [ht, Set.inter_univ, measure_univ, mul_one] Qed
ProbabilityTheory.kernel.indepSet_empty_right α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : Set Ω ⊢ IndepSet s ∅ κ simp only [IndepSet, generateFrom_singleton_empty] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : Set Ω ⊢ Indep (generateFrom {s}) ⊥ κ exact indep_bot_right _ ** Qed
ProbabilityTheory.kernel.IndepSets.union α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ ⊢ IndepSets (s₁ ∪ s₂) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ t1 t2 : Set Ω ht1 : t1 ∈ s₁ ∪ s₂ ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 cases' (Set.mem_union _ _ _).mp ht1 with ht1₁ ht1₂ case inl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ t1 t2 : Set Ω ht1 : t1 ∈ s₁ ∪ s₂ ht2 : t2 ∈ s' ht1₁ : t1 ∈ s₁ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h₁ t1 t2 ht1₁ ht2 case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s₁ s₂ s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h₁ : IndepSets s₁ s' κ h₂ : IndepSets s₂ s' κ t1 t2 : Set Ω ht1 : t1 ∈ s₁ ∪ s₂ ht2 : t2 ∈ s' ht1₂ : t1 ∈ s₂ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h₂ t1 t2 ht1₂ ht2 Qed
ProbabilityTheory.kernel.IndepSets.iUnion α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ ⊢ IndepSets (⋃ n, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋃ n, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rw [Set.mem_iUnion] at ht1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : ∃ i, t1 ∈ s i ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 cases' ht1 with n ht1 case intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α hyp : ∀ (n : ι), IndepSets (s n) s' κ t1 t2 : Set Ω ht2 : t2 ∈ s' n : ι ht1 : t1 ∈ s n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact hyp n t1 t2 ht1 ht2 Qed
ProbabilityTheory.kernel.IndepSets.bUnion α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ ⊢ IndepSets (⋃ n ∈ u, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋃ n ∈ u, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 simp_rw [Set.mem_iUnion] at ht1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ t1 t2 : Set Ω ht2 : t2 ∈ s' ht1 : ∃ i i_1, t1 ∈ s i ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rcases ht1 with ⟨n, hpn, ht1⟩ case intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι hyp : ∀ (n : ι), n ∈ u → IndepSets (s n) s' κ t1 t2 : Set Ω ht2 : t2 ∈ s' n : ι hpn : n ∈ u ht1 : t1 ∈ s n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact hyp n hpn t1 t2 ht1 ht2 Qed
ProbabilityTheory.kernel.IndepSets.iInter α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h : ∃ n, IndepSets (s n) s' κ ⊢ IndepSets (⋂ n, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h : ∃ n, IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 cases' h with n h case intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n, s n ht2 : t2 ∈ s' n : ι h : IndepSets (s n) s' κ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h t1 t2 (Set.mem_iInter.mp ht1 n) ht2 Qed
ProbabilityTheory.kernel.IndepSets.bInter α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι h : ∃ n, n ∈ u ∧ IndepSets (s n) s' κ ⊢ IndepSets (⋂ n ∈ u, s n) s' κ intro t1 t2 ht1 ht2 ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι h : ∃ n, n ∈ u ∧ IndepSets (s n) s' κ t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n ∈ u, s n ht2 : t2 ∈ s' ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rcases h with ⟨n, hn, h⟩ case intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s : ι → Set (Set Ω) s' : Set (Set Ω) _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α u : Set ι t1 t2 : Set Ω ht1 : t1 ∈ ⋂ n ∈ u, s n ht2 : t2 ∈ s' n : ι hn : n ∈ u h : IndepSets (s n) s' κ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 exact h t1 t2 (Set.biInter_subset_of_mem hn ht1) ht2 Qed
ProbabilityTheory.kernel.indepSets_singleton_iff ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α s t : Set Ω _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s ∩ t) = ↑↑(↑κ a) s * ↑↑(↑κ a) t s1 t1 : Set Ω hs1 : s1 ∈ {s} ht1 : t1 ∈ {t} ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (s1 ∩ t1) = ↑↑(↑κ a) s1 * ↑↑(↑κ a) t1 rwa [Set.mem_singleton_iff.mp hs1, Set.mem_singleton_iff.mp ht1] Qed
ProbabilityTheory.kernel.iIndep.indep α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m : ι → MeasurableSpace Ω _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : iIndep m κ i j : ι hij : i ≠ j ⊢ Indep (m i) (m j) κ change IndepSets ((fun x => MeasurableSet[m x]) i) ((fun x => MeasurableSet[m x]) j) κ μ α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m : ι → MeasurableSpace Ω _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : iIndep m κ i j : ι hij : i ≠ j ⊢ IndepSets ((fun x => MeasurableSet) i) ((fun x => MeasurableSet) j) κ exact iIndepSets.indepSets h_indep hij ** Qed
ProbabilityTheory.kernel.IndepSets.indep_aux ** α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 ** refine @induction_on_inter _ (fun t ↦ ∀ᵐ a ∂μ, κ a (t1 ∩ t) = κ a t1 * κ a t) _ m₂ hpm2 hp2 ?_ ?_ ?_ ?_ t2 ht2m ** case refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) ∅ simp only [Set.inter_empty, measure_empty, mul_zero, eq_self_iff_true, Filter.eventually_true] case refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ (t : Set Ω), t ∈ p2 → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) t exact fun t ht_mem_p2 ↦ hyp t1 t ht1 ht_mem_p2 case refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ (t : Set Ω), MeasurableSet t → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) t → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) tᶜ intros t ht h case refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ tᶜ) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) tᶜ filter_upwards [h] with a ha case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t a : α ha : ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t ⊢ ↑↑(↑κ a) (t1 ∩ tᶜ) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) tᶜ have : t1 ∩ tᶜ = t1 \ (t1 ∩ t) := by rw [Set.diff_self_inter, Set.diff_eq_compl_inter, Set.inter_comm] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t a : α ha : ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t this : t1 ∩ tᶜ = t1 \ (t1 ∩ t) ⊢ ↑↑(↑κ a) (t1 ∩ tᶜ) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) tᶜ rw [this, measure_diff (Set.inter_subset_left _ _) (ht1m.inter (h2 _ ht)) (measure_ne_top (κ a) _), measure_compl (h2 _ ht) (measure_ne_top (κ a) t), measure_univ, ENNReal.mul_sub (fun _ _ ↦ measure_ne_top (κ a) _), mul_one, ha] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 t : Set Ω ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t a : α ha : ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t ⊢ t1 ∩ tᶜ = t1 \ (t1 ∩ t) rw [Set.diff_self_inter, Set.diff_eq_compl_inter, Set.inter_comm] case refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 ⊢ ∀ (f : ℕ → Set Ω), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) (f i)) → (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) (⋃ i, f i) intros f hf_disj hf_meas h case refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ (i : ℕ), (fun t => ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t) (f i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ ⋃ i, f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) rw [← ae_all_iff] at h case refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ ⋃ i, f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) filter_upwards [h] with a ha case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ↑↑(↑κ a) (t1 ∩ ⋃ i, f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) rw [Set.inter_iUnion, measure_iUnion] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∑' (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (⋃ i, f i) rw [measure_iUnion hf_disj (fun i ↦ h2 _ (hf_meas i))] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∑' (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ∑' (i : ℕ), ↑↑(↑κ a) (f i) rw [← ENNReal.tsum_mul_left] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∑' (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ∑' (i : ℕ), ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) congr with i case h.e_f.h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) i : ℕ ⊢ ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) rw [ha i] case h.hn α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ Pairwise (Disjoint on fun i => t1 ∩ f i) intros i j hij case h.hn α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) i j : ℕ hij : i ≠ j ⊢ (Disjoint on fun i => t1 ∩ f i) i j rw [Function.onFun, Set.inter_comm t1, Set.inter_comm t1] case h.hn α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) i j : ℕ hij : i ≠ j ⊢ Disjoint (f i ∩ t1) (f j ∩ t1) exact Disjoint.inter_left _ (Disjoint.inter_right _ (hf_disj hij)) case h.h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α m₂ m : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ p1 p2 : Set (Set Ω) h2 : m₂ ≤ m hp2 : IsPiSystem p2 hpm2 : m₂ = generateFrom p2 hyp : IndepSets p1 p2 κ t1 t2 : Set Ω ht1 : t1 ∈ p1 ht1m : MeasurableSet t1 ht2m : MeasurableSet t2 f : ℕ → Set Ω hf_disj : Pairwise (Disjoint on f) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) a : α ha : ∀ (i : ℕ), ↑↑(↑κ a) (t1 ∩ f i) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) (f i) ⊢ ∀ (i : ℕ), MeasurableSet (t1 ∩ f i) exact fun i ↦ ht1m.inter (h2 _ (hf_meas i)) Qed
ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_of_disjoint α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ Indep (generateFrom {t \| ∃ n, n ∈ S ∧ s n = t}) (generateFrom {t \| ∃ k, k ∈ T ∧ s k = t}) κ rw [← generateFrom_piiUnionInter_singleton_left, ← generateFrom_piiUnionInter_singleton_left] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ Indep (generateFrom (piiUnionInter (fun k => {s k}) S)) (generateFrom (piiUnionInter (fun k => {s k}) T)) κ refine' IndepSets.indep' (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurableSet_generateFrom ht)) (fun t ht => generateFrom_piiUnionInter_le _ _ _ _ (measurableSet_generateFrom ht)) _ _ _ case refine'_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T t : Set Ω ht : t ∈ piiUnionInter (fun k => {s k}) S ⊢ ∀ (n : ι), generateFrom {s n} ≤ _mΩ exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k case refine'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T t : Set Ω ht : t ∈ piiUnionInter (fun k => {s k}) T ⊢ ∀ (n : ι), generateFrom {s n} ≤ _mΩ exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k case refine'_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun k => {s k}) S) exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _ case refine'_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun k => {s k}) T) exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _ case refine'_5 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun k => {s k}) S) (piiUnionInter (fun k => {s k}) T) κ classical exact indepSets_piiUnionInter_of_disjoint (iIndep.iIndepSets (fun n => rfl) hs) hST case refine'_5 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun k => {s k}) S) (piiUnionInter (fun k => {s k}) T) κ exact indepSets_piiUnionInter_of_disjoint (iIndep.iIndepSets (fun n => rfl) hs) hST ** Qed
ProbabilityTheory.kernel.indep_iSup_of_disjoint α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) κ refine' IndepSets.indep (iSup₂_le fun i _ => h_le i) (iSup₂_le fun i _ => h_le i) _ _ (generateFrom_piiUnionInter_measurableSet m S).symm (generateFrom_piiUnionInter_measurableSet m T).symm _ case refine'_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun n => {s \| MeasurableSet s}) S) exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _ case refine'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IsPiSystem (piiUnionInter (fun n => {s \| MeasurableSet s}) T) exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _ case refine'_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun n => {s \| MeasurableSet s}) S) (piiUnionInter (fun n => {s \| MeasurableSet s}) T) κ classical exact indepSets_piiUnionInter_of_disjoint h_indep hST case refine'_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ m : ι → MeasurableSpace Ω h_le : ∀ (i : ι), m i ≤ _mΩ h_indep : iIndep m κ S T : Set ι hST : Disjoint S T ⊢ IndepSets (piiUnionInter (fun n => {s \| MeasurableSet s}) S) (piiUnionInter (fun n => {s \| MeasurableSet s}) T) κ exact indepSets_piiUnionInter_of_disjoint h_indep hST ** Qed
ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_lt α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : Preorder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i : ι ⊢ Indep (generateFrom {s i}) (generateFrom {t \| ∃ j, j < i ∧ s j = t}) κ convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {i} { j \| j < i } (Set.disjoint_singleton_left.mpr (lt_irrefl _)) case h.e'_4.h.e'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : Preorder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i : ι ⊢ {s i} = {t \| ∃ n, n ∈ {i} ∧ s n = t} simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton'] ** Qed
ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_le α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : LinearOrder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i k : ι hk : i < k ⊢ Indep (generateFrom {s k}) (generateFrom {t \| ∃ j, j ≤ i ∧ s j = t}) κ convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {k} { j \| j ≤ i } (Set.disjoint_singleton_left.mpr hk.not_le) case h.e'_4.h.e'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝¹ : LinearOrder ι inst✝ : IsMarkovKernel κ s : ι → Set Ω hsm : ∀ (n : ι), MeasurableSet (s n) hs : iIndepSet s κ i k : ι hk : i < k ⊢ {s k} = {t \| ∃ n, n ∈ {k} ∧ s n = t} simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton'] ** Qed
ProbabilityTheory.kernel.iIndepSets.piiUnionInter_of_not_mem α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S ⊢ IndepSets (piiUnionInter π ↑S) (π a) κ rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia ** case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : ↑s ⊆ ↑S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 rw [Finset.coe_subset] at hs_mem case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ) case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS]) case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_μ_t1 : ∀ᵐ a' ∂μ, κ a' t1 = ∏ n in s, κ a' (f n) := by filter_upwards [hp_ind s h_f_mem_pi] with a' ha' rw [h_t1, ← ha'] case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_t2 : t2 = f a := by simp case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have h_μ_inter : ∀ᵐ a' ∂μ, κ a' (t1 ∩ t2) = ∏ n in insert a s, κ a' (f n) := by have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm] filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha' rw [h_t1_inter_t2, ← ha'] case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_μ_inter : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 have has : a ∉ s := fun has_mem => haS (hs_mem has_mem) case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_μ_inter : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) has : ¬a ∈ s ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (t1 ∩ t2) = ↑↑(↑κ a) t1 * ↑↑(↑κ a) t2 filter_upwards [h_μ_t1, h_μ_inter] with a' ha1 ha2 case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_μ_inter : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) has : ¬a ∈ s a' : α ha1 : ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) ha2 : ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) ⊢ ↑↑(↑κ a') (t1 ∩ t2) = ↑↑(↑κ a') t1 * ↑↑(↑κ a') t2 rw [ha2, Finset.prod_insert has, h_t2, mul_comm, ha1] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ ⊢ ∀ (n : ι), n ∈ insert a s → f n ∈ π n intro n hn_mem_insert α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s ⊢ f n ∈ π n dsimp only α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem case inl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n = a ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n simp [hn_mem, ht2_mem_pia] case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n ∈ s ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem) case inr α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n ∈ s hn_ne_a : n ≠ a ⊢ (if n = a then t2 else if n ∈ s then ft1 n else Set.univ) ∈ π n simp [hn_ne_a, hn_mem, hft1_mem n hn_mem] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ n : ι hn_mem_insert : n ∈ insert a s hn_mem : n ∈ s ⊢ n ≠ a rintro rfl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) S : Finset ι hp_ind : iIndepSets π κ t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x n : ι hn_mem : n ∈ s haS : ¬n ∈ S ht2_mem_pia : t2 ∈ π n f : ι → Set Ω := fun n_1 => if n_1 = n then t2 else if n_1 ∈ s then ft1 n_1 else Set.univ hn_mem_insert : n ∈ insert n s ⊢ False exact haS (hs_mem hn_mem) α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n x : ι hxS : x ∈ s ⊢ x ∈ insert a s simp [hxS] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n ⊢ t1 = ⋂ n ∈ s, f n suffices h_forall : ∀ n ∈ s, f n = ft1 n case h_forall α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n ⊢ ∀ (n : ι), n ∈ s → f n = ft1 n intro n hnS case h_forall α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n n : ι hnS : n ∈ s ⊢ f n = ft1 n have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS) case h_forall α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n n : ι hnS : n ∈ s hn_ne_a : n ≠ a ⊢ f n = ft1 n simp_rw [if_pos hnS, if_neg hn_ne_a] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n ⊢ t1 = ⋂ n ∈ s, f n rw [ht1_eq] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n ⊢ ⋂ x ∈ s, ft1 x = ⋂ n ∈ s, f n ext x case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n x : Ω ⊢ x ∈ ⋂ x ∈ s, ft1 x ↔ x ∈ ⋂ n ∈ s, f n simp_rw [Set.mem_iInter] case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_forall : ∀ (n : ι), n ∈ s → f n = ft1 n x : Ω ⊢ (∀ (i : ι), i ∈ s → x ∈ ft1 i) ↔ ∀ (i : ι), i ∈ s → x ∈ if i = a then t2 else if i ∈ s then ft1 i else Set.univ conv => lhs; intro i hns; rw [← h_forall i hns] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n n : ι hnS : n ∈ s ⊢ n ≠ a rintro rfl α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) S : Finset ι hp_ind : iIndepSets π κ t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x n : ι hnS : n ∈ s haS : ¬n ∈ S ht2_mem_pia : t2 ∈ π n f : ι → Set Ω := fun n_1 => if n_1 = n then t2 else if n_1 ∈ s then ft1 n_1 else Set.univ h_f_mem : ∀ (n_1 : ι), n_1 ∈ insert n s → f n_1 ∈ π n_1 h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n ⊢ False exact haS (hs_mem hnS) α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n ⊢ ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) filter_upwards [hp_ind s h_f_mem_pi] with a' ha' case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n a' : α ha' : ↑↑(↑κ a') (⋂ i ∈ s, f i) = ∏ i in s, ↑↑(↑κ a') (f i) ⊢ ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) rw [h_t1, ← ha'] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) ⊢ t2 = f a simp α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a ⊢ ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n ⊢ ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha' case h α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n a' : α ha' : ↑↑(↑κ a') (⋂ i ∈ insert a s, f i) = ∏ i in insert a s, ↑↑(↑κ a') (f i) ⊢ ↑↑(↑κ a') (t1 ∩ t2) = ∏ n in insert a s, ↑↑(↑κ a') (f n) rw [h_t1_inter_t2, ← ha'] α : Type u_1 Ω : Type u_2 ι : Type u_3 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α π : ι → Set (Set Ω) a : ι S : Finset ι hp_ind : iIndepSets π κ haS : ¬a ∈ S t1 t2 : Set Ω s : Finset ι hs_mem : s ⊆ S ft1 : ι → Set Ω hft1_mem : ∀ (x : ι), x ∈ s → ft1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ s, ft1 x ht2_mem_pia : t2 ∈ π a f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else Set.univ h_f_mem : ∀ (n : ι), n ∈ insert a s → f n ∈ π n h_f_mem_pi : ∀ (n : ι), n ∈ s → f n ∈ π n h_t1 : t1 = ⋂ n ∈ s, f n h_μ_t1 : ∀ᵐ (a' : α) ∂μ, ↑↑(↑κ a') t1 = ∏ n in s, ↑↑(↑κ a') (f n) h_t2 : t2 = f a ⊢ t1 ∩ t2 = ⋂ n ∈ insert a s, f n rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm] Qed
ProbabilityTheory.kernel.indepSet_iff_indepSets_singleton α : Type u_1 Ω : Type u_2 ι : Type u_3 s t : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m0 : MeasurableSpace Ω hs_meas : MeasurableSet s ht_meas : MeasurableSet t κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ h : IndepSets {s} {t} κ u : Set Ω hu : u ∈ {s} ⊢ MeasurableSet u rwa [Set.mem_singleton_iff.mp hu] α : Type u_1 Ω : Type u_2 ι : Type u_3 s t : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m0 : MeasurableSpace Ω hs_meas : MeasurableSet s ht_meas : MeasurableSet t κ : { x // x ∈ kernel α Ω } μ : Measure α inst✝ : IsMarkovKernel κ h : IndepSets {s} {t} κ u : Set Ω hu : u ∈ {t} ⊢ MeasurableSet u rwa [Set.mem_singleton_iff.mp hu] ** Qed
ProbabilityTheory.kernel.Indep.indepSet_of_measurableSet α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t ⊢ IndepSet s t κ refine fun s' t' hs' ht' => h_indep s' t' ?_ ?_ case refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ s' ∈ {s \| MeasurableSet s} refine @generateFrom_induction _ (fun u => MeasurableSet[m₁] u) {s} ?_ ?_ ?_ ?_ _ hs' case refine_1.refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t : Set Ω), t ∈ {s} → (fun u => MeasurableSet u) t simp only [Set.mem_singleton_iff, forall_eq, hs] case refine_1.refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ (fun u => MeasurableSet u) ∅ exact @MeasurableSet.empty _ m₁ case refine_1.refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t : Set Ω), (fun u => MeasurableSet u) t → (fun u => MeasurableSet u) tᶜ exact fun u hu => hu.compl case refine_1.refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (f : ℕ → Set Ω), (∀ (n : ℕ), (fun u => MeasurableSet u) (f n)) → (fun u => MeasurableSet u) (⋃ i, f i) exact fun f hf => MeasurableSet.iUnion hf case refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ t' ∈ {s \| MeasurableSet s} refine @generateFrom_induction _ (fun u => MeasurableSet[m₂] u) {t} ?_ ?_ ?_ ?_ _ ht' case refine_2.refine_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t_1 : Set Ω), t_1 ∈ {t} → (fun u => MeasurableSet u) t_1 simp only [Set.mem_singleton_iff, forall_eq, ht] case refine_2.refine_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ (fun u => MeasurableSet u) ∅ exact @MeasurableSet.empty _ m₂ case refine_2.refine_3 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (t : Set Ω), (fun u => MeasurableSet u) t → (fun u => MeasurableSet u) tᶜ exact fun u hu => hu.compl case refine_2.refine_4 α : Type u_1 Ω : Type u_2 ι : Type u_3 s✝ t✝ : Set Ω S T : Set (Set Ω) _mα : MeasurableSpace α m₁ m₂ m0 : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α h_indep : Indep m₁ m₂ κ s t : Set Ω hs : MeasurableSet s ht : MeasurableSet t s' t' : Set Ω hs' : s' ∈ {s_1 \| MeasurableSet s_1} ht' : t' ∈ {s \| MeasurableSet s} ⊢ ∀ (f : ℕ → Set Ω), (∀ (n : ℕ), (fun u => MeasurableSet u) (f n)) → (fun u => MeasurableSet u) (⋃ i, f i) exact fun f hf => MeasurableSet.iUnion hf ** Qed
ProbabilityTheory.kernel.indepFun_iff_measure_inter_preimage_eq_mul ** α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' ⊢ IndepFun f g κ ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) constructor <;> intro h case mp α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' h : IndepFun f g κ ⊢ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) refine' fun s t hs ht => h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩ case mpr α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' h : ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) ⊢ IndepFun f g κ rintro _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩ case mpr.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' h : ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) s : Set β hs : MeasurableSet s t : Set β' ht : MeasurableSet t ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) exact h s t hs ht Qed
ProbabilityTheory.kernel.iIndepFun_iff_measure_inter_preimage_eq_mul α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i ⊢ iIndepFun m f κ ↔ ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) refine' ⟨fun h S sets h_meas => h _ fun i hi_mem => ⟨sets i, h_meas i hi_mem, rfl⟩, _⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i ⊢ (∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i)) → iIndepFun m f κ intro h S setsΩ h_meas α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) let setsβ : ∀ i : ι, Set (β i) := fun i => dite (i ∈ S) (fun hi_mem => (h_meas i hi_mem).choose) fun _ => Set.univ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) have h_measβ : ∀ i ∈ S, MeasurableSet[m i] (setsβ i) := by intro i hi_mem simp_rw [dif_pos hi_mem] exact (h_meas i hi_mem).choose_spec.1 α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) have h_preim : ∀ i ∈ S, setsΩ i = f i ⁻¹' setsβ i := by intro i hi_mem simp_rw [dif_pos hi_mem] exact (h_meas i hi_mem).choose_spec.2.symm α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) have h_right_eq : ∀ a, (∏ i in S, κ a (setsΩ i)) = ∏ i in S, κ a ((f i) ⁻¹' (setsβ i)) := by refine' fun a ↦ Finset.prod_congr rfl fun i hi_mem => _ rw [h_preim i hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) h_right_eq : ∀ (a : α), ∏ i in S, ↑↑(↑κ a) (setsΩ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) filter_upwards [h S h_measβ] with a ha case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) h_right_eq : ∀ (a : α), ∏ i in S, ↑↑(↑κ a) (setsΩ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) a : α ha : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) ⊢ ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ∏ i in S, ↑↑(↑κ a) (setsΩ i) rw [h_left_eq a, h_right_eq a, ha] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ ⊢ ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) intro i hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ i : ι hi_mem : i ∈ S ⊢ MeasurableSet (setsβ i) simp_rw [dif_pos hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ i : ι hi_mem : i ∈ S ⊢ MeasurableSet (Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i)) exact (h_meas i hi_mem).choose_spec.1 α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) ⊢ ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i intro i hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) i : ι hi_mem : i ∈ S ⊢ setsΩ i = f i ⁻¹' setsβ i simp_rw [dif_pos hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) i : ι hi_mem : i ∈ S ⊢ setsΩ i = f i ⁻¹' Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) exact (h_meas i hi_mem).choose_spec.2.symm α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i ⊢ ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) intro a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α ⊢ ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) congr with x case e_a.h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω ⊢ x ∈ ⋂ i ∈ S, setsΩ i ↔ x ∈ ⋂ i ∈ S, f i ⁻¹' setsβ i simp_rw [Set.mem_iInter] case e_a.h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω ⊢ (∀ (i : ι), i ∈ S → x ∈ setsΩ i) ↔ ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if h : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ constructor <;> intro h i hi_mem <;> specialize h i hi_mem case e_a.h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h✝ : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω i : ι hi_mem : i ∈ S h : x ∈ setsΩ i ⊢ x ∈ f i ⁻¹' if h : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ rwa [h_preim i hi_mem] at h case e_a.h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h✝ : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i a : α x : Ω i : ι hi_mem : i ∈ S h : x ∈ f i ⁻¹' if h : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ ⊢ x ∈ setsΩ i rwa [h_preim i hi_mem] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) ⊢ ∀ (a : α), ∏ i in S, ↑↑(↑κ a) (setsΩ i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' setsβ i) refine' fun a ↦ Finset.prod_congr rfl fun i hi_mem => _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' ι : Type u_8 β : ι → Type u_9 m : (x : ι) → MeasurableSpace (β x) f : (i : ι) → Ω → β i h : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) S : Finset ι setsΩ : ι → Set Ω h_meas : ∀ (i : ι), i ∈ S → setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i setsβ : (i : ι) → Set (β i) := fun i => if hi_mem : i ∈ S then Exists.choose (_ : setsΩ i ∈ (fun x => {s \| MeasurableSet s}) i) else Set.univ h_measβ : ∀ (i : ι), i ∈ S → MeasurableSet (setsβ i) h_preim : ∀ (i : ι), i ∈ S → setsΩ i = f i ⁻¹' setsβ i h_left_eq : ∀ (a : α), ↑↑(↑κ a) (⋂ i ∈ S, setsΩ i) = ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' setsβ i) a : α i : ι hi_mem : i ∈ S ⊢ ↑↑(↑κ a) (setsΩ i) = ↑↑(↑κ a) (f i ⁻¹' setsβ i) rw [h_preim i hi_mem] ** Qed
ProbabilityTheory.kernel.indepFun_iff_indepSet_preimage α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g ⊢ IndepFun f g κ ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ refine' indepFun_iff_measure_inter_preimage_eq_mul.trans _ ** α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g ⊢ (∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t)) ↔ ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ constructor <;> intro h s t hs ht <;> specialize h s t hs ht case mp α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g s : Set β t : Set β' hs : MeasurableSet s ht : MeasurableSet t h : ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) ⊢ IndepSet (f ⁻¹' s) (g ⁻¹' t) κ rwa [indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) κ μ] case mpr α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' inst✝ : IsMarkovKernel κ hf : Measurable f hg : Measurable g s : Set β t : Set β' hs : MeasurableSet s ht : MeasurableSet t h : IndepSet (f ⁻¹' s) (g ⁻¹' t) κ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑(↑κ a) (f ⁻¹' s) * ↑↑(↑κ a) (g ⁻¹' t) rwa [← indepSet_iff_measure_inter_eq_mul (hf hs) (hg ht) κ μ] Qed
ProbabilityTheory.kernel.IndepFun.ae_eq α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' ⊢ IndepFun f' g' κ rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) filter_upwards [hf, hg, hfg _ _ ⟨_, hA, rfl⟩ ⟨_, hB, rfl⟩] with a hf' hg' hfg' case h α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B a : α hf' : f =ᵐ[↑κ a] f' hg' : g =ᵐ[↑κ a] g' hfg' : ↑↑(↑κ a) (f ⁻¹' A ∩ g ⁻¹' B) = ↑↑(↑κ a) (f ⁻¹' A) * ↑↑(↑κ a) (g ⁻¹' B) ⊢ ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) have h1 : f ⁻¹' A =ᵐ[κ a] f' ⁻¹' A := hf'.fun_comp A case h α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B a : α hf' : f =ᵐ[↑κ a] f' hg' : g =ᵐ[↑κ a] g' hfg' : ↑↑(↑κ a) (f ⁻¹' A ∩ g ⁻¹' B) = ↑↑(↑κ a) (f ⁻¹' A) * ↑↑(↑κ a) (g ⁻¹' B) h1 : f ⁻¹' A =ᵐ[↑κ a] f' ⁻¹' A ⊢ ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) have h2 : g ⁻¹' B =ᵐ[κ a] g' ⁻¹' B := hg'.fun_comp B case h α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β g✝ : Ω → β' mβ : MeasurableSpace β f g f' g' : Ω → β hfg : IndepFun f g κ hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[↑κ a] f' hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[↑κ a] g' A : Set β hA : MeasurableSet A B : Set β hB : MeasurableSet B a : α hf' : f =ᵐ[↑κ a] f' hg' : g =ᵐ[↑κ a] g' hfg' : ↑↑(↑κ a) (f ⁻¹' A ∩ g ⁻¹' B) = ↑↑(↑κ a) (f ⁻¹' A) * ↑↑(↑κ a) (g ⁻¹' B) h1 : f ⁻¹' A =ᵐ[↑κ a] f' ⁻¹' A h2 : g ⁻¹' B =ᵐ[↑κ a] g' ⁻¹' B ⊢ ↑↑(↑κ a) (f' ⁻¹' A ∩ g' ⁻¹' B) = ↑↑(↑κ a) (f' ⁻¹' A) * ↑↑(↑κ a) (g' ⁻¹' B) rwa [← measure_congr h1, ← measure_congr h2, ← measure_congr (h1.inter h2)] Qed
ProbabilityTheory.kernel.IndepFun.comp α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ ⊢ IndepFun (φ ∘ f) (ψ ∘ g) κ rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ A : Set γ hA : MeasurableSet A B : Set γ' hB : MeasurableSet B ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (φ ∘ f ⁻¹' A ∩ ψ ∘ g ⁻¹' B) = ↑↑(↑κ a) (φ ∘ f ⁻¹' A) * ↑↑(↑κ a) (ψ ∘ g ⁻¹' B) apply hfg case intro.intro.intro.intro.a α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ A : Set γ hA : MeasurableSet A B : Set γ' hB : MeasurableSet B ⊢ φ ∘ f ⁻¹' A ∈ {s \| MeasurableSet s} exact ⟨φ ⁻¹' A, hφ hA, Set.preimage_comp.symm⟩ case intro.intro.intro.intro.a α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' mβ : MeasurableSpace β mβ' : MeasurableSpace β' mγ : MeasurableSpace γ mγ' : MeasurableSpace γ' φ : β → γ ψ : β' → γ' hfg : IndepFun f g κ hφ : Measurable φ hψ : Measurable ψ A : Set γ hA : MeasurableSet A B : Set γ' hB : MeasurableSet B ⊢ ψ ∘ g ⁻¹' B ∈ {s \| MeasurableSet s} exact ⟨ψ ⁻¹' B, hψ hB, Set.preimage_comp.symm⟩ Qed
ProbabilityTheory.kernel.iIndepFun.indepFun_finset α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πSβ := Set.pi (Set.univ : Set S) '' Set.pi (Set.univ : Set S) fun i => { s : Set (β i) \| MeasurableSet[m i] s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πS := { s : Set Ω \| ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπS_pi : IsPiSystem πS := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπS_gen : (MeasurableSpace.pi.comap fun a (i : S) => f i a) = generateFrom πS := by rw [generateFrom_pi.symm, comap_generateFrom] congr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πTβ := Set.pi (Set.univ : Set T) '' Set.pi (Set.univ : Set T) fun i => { s : Set (β i) \| MeasurableSet[m i] s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ let πT := { s : Set Ω \| ∃ t ∈ πTβ, (fun a (i : T) => f i a) ⁻¹' t = s } α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπT_pi : IsPiSystem πT := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ have hπT_gen : (MeasurableSpace.pi.comap fun a (i : T) => f i a) = generateFrom πT := by rw [generateFrom_pi.symm, comap_generateFrom] congr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT ⊢ IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) κ refine IndepSets.indep (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) hπS_pi hπT_pi hπS_gen hπT_gen ?_ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT ⊢ IndepSets πS πT κ rintro _ _ ⟨s, ⟨sets_s, hs1, hs2⟩, rfl⟩ ⟨t, ⟨sets_t, ht1, ht2⟩, rfl⟩ ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs1 : sets_s ∈ Set.pi Set.univ fun i => {s \| MeasurableSet s} hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht1 : sets_t ∈ Set.pi Set.univ fun i => {s \| MeasurableSet s} ht2 : Set.pi Set.univ sets_t = t ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s ∩ (fun a i => f (↑i) a) ⁻¹' t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' t) simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1 case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s ∩ (fun a i => f (↑i) a) ⁻¹' t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' t) rw [← hs2, ← ht2] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IsPiSystem πS exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS ⊢ MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS rw [generateFrom_pi.symm, comap_generateFrom] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS ⊢ generateFrom ((Set.preimage fun a i => f (↑i) a) '' (Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s})) = generateFrom πS congr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} ⊢ IsPiSystem πT exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT ⊢ MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT rw [generateFrom_pi.symm, comap_generateFrom] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT ⊢ generateFrom ((Set.preimage fun a i => f (↑i) a) '' (Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s})) = generateFrom πT congr case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) let sets_s' : ∀ i : ι, Set (β i) := fun i => dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by intro i hi; simp_rw [dif_pos hi] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_sets_s'_univ : ∀ {i} (_hi : i ∈ T), sets_s' i = Set.univ := by intro i hi; simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) let sets_t' : ∀ i : ι, Set (β i) := fun i => dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_sets_t'_univ : ∀ {i} (_hi : i ∈ S), sets_t' i = Set.univ := by intro i hi; simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by intro i hi; rw [h_sets_s'_eq hi]; exact hs1 _ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by intro i hi; simp_rw [dif_pos hi]; exact ht1 _ case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) rw [iIndepFun_iff_measure_inter_preimage_eq_mul] at hf_Indep case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) filter_upwards [hf_Indep S h_meas_s', hf_Indep T h_meas_t', hf_Indep (S ∪ T) h_meas_inter] with a h_indepS h_indepT h_indepST case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) = ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑(↑κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) rw [h_eq_inter_S, h_eq_inter_T, h_indepS, h_indepT, h_Inter_inter, h_indepST, Finset.prod_union hST] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ (∏ x in S, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x))) * ∏ x in T, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = (∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i)) * ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) congr 1 α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ⊢ sets_s' i = sets_s { val := i, property := hi } simp_rw [dif_pos hi] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } ⊢ ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } i : ι hi : i ∈ T ⊢ sets_s' i = Set.univ simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ⊢ sets_t' i = Set.univ simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ ⊢ ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ i : ι hi : i ∈ S ⊢ MeasurableSet (sets_s' i) rw [h_sets_s'_eq hi] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ i : ι hi : i ∈ S ⊢ MeasurableSet (sets_s { val := i, property := hi }) exact hs1 _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) ⊢ ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) intro i hi α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) i : ι hi : i ∈ T ⊢ MeasurableSet (sets_t' i) simp_rw [dif_pos hi] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) i : ι hi : i ∈ T ⊢ MeasurableSet (sets_t { val := i, property := hi }) exact ht1 _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) ⊢ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i ext1 x case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω ⊢ x ∈ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s ↔ x ∈ ⋂ i ∈ S, f i ⁻¹' sets_s' i simp_rw [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω ⊢ (∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i) ↔ ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ constructor <;> intro h case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i ⊢ ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ intro i hi case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i i : ι hi : i ∈ S ⊢ x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ simp only [h_sets_s'_eq hi, Set.mem_preimage] case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i i : ι hi : i ∈ S ⊢ f i x ∈ sets_s { val := i, property := hi } exact h ⟨i, hi⟩ case h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i rintro ⟨i, hi⟩ case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω h : ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ⊢ f (↑{ val := i, property := hi }) x ∈ sets_s { val := i, property := hi } specialize h i hi case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) x : Ω i : ι hi : i ∈ S h : x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f (↑{ val := i, property := hi }) x ∈ sets_s { val := i, property := hi } rwa [dif_pos hi] at h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i ⊢ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i ext1 x case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω ⊢ x ∈ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t ↔ x ∈ ⋂ i ∈ T, f i ⁻¹' sets_t' i simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω ⊢ (∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i) ↔ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ constructor <;> intro h case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i ⊢ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ intro i hi case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i i : ι hi : i ∈ T ⊢ f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ simp_rw [dif_pos hi] case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i i : ι hi : i ∈ T ⊢ f i x ∈ sets_t { val := i, property := hi } exact h ⟨i, hi⟩ case h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i rintro ⟨i, hi⟩ case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω h : ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hi : i ∈ T ⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi } specialize h i hi case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω i : ι hi : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi } simp_rw [dif_pos hi] at h case h.mpr.mk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i x : Ω i : ι hi : i ∈ T h : f i x ∈ sets_t { val := i, property := hi } ⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi } exact h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i ⊢ (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) ext1 x case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω ⊢ x ∈ (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i ↔ x ∈ ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) simp_rw [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω ⊢ ((∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ) ↔ ∀ (i : ι), i ∈ S ∨ i ∈ T → f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ constructor <;> intro h case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ ∀ (i : ι), i ∈ S ∨ i ∈ T → f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ intro i hi case h.mp α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hi : i ∈ S ∨ i ∈ T ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ cases' hi with hiS hiT case h.mp.inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hiS : i ∈ S ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ replace h := h.1 i hiS case h.mp.inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiS : i ∈ S h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ simp_rw [dif_pos hiS, dif_neg (Finset.disjoint_left.mp hST hiS)] case h.mp.inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiS : i ∈ S h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f i x ∈ sets_s { val := i, property := hiS } ∩ Set.univ exact ⟨by rwa [dif_pos hiS] at h, Set.mem_univ _⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiS : i ∈ S h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ ⊢ f i x ∈ sets_s { val := i, property := hiS } rwa [dif_pos hiS] at h case h.mp.inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ i : ι hiT : i ∈ T ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ replace h := h.2 i hiT case h.mp.inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiT : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ simp_rw [dif_pos hiT, dif_neg (Finset.disjoint_right.mp hST hiT)] case h.mp.inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiT : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f i x ∈ Set.univ ∩ sets_t { val := i, property := hiT } exact ⟨Set.mem_univ _, by rwa [dif_pos hiT] at h⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω i : ι hiT : i ∈ T h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ f i x ∈ sets_t { val := i, property := hiT } rwa [dif_pos hiT] at h case h.mpr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i x : Ω h : ∀ (i : ι), i ∈ S ∨ i ∈ T → f i x ∈ (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ ⊢ (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) ⊢ ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) intros i hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ∪ T ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) rw [Finset.mem_union] at hi_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ∨ i ∈ T ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) cases' hi_mem with hi_mem hi_mem case inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) rw [h_sets_t'_univ hi_mem, Set.inter_univ] case inl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ S ⊢ MeasurableSet (sets_s' i) exact h_meas_s' i hi_mem case inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ T ⊢ MeasurableSet (sets_s' i ∩ sets_t' i) rw [h_sets_s'_univ hi_mem, Set.univ_inter] case inr α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) i : ι hi_mem : i ∈ T ⊢ MeasurableSet (sets_t' i) exact h_meas_t' i hi_mem case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ ∏ x in S, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) refine' Finset.prod_congr rfl fun i hi => _ case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) i : ι hi : i ∈ S ⊢ ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ↑↑(↑κ a) (f i ⁻¹' sets_s' i) rw [h_sets_t'_univ hi, Set.inter_univ] case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) ⊢ ∏ x in T, ↑↑(↑κ a) (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) refine' Finset.prod_congr rfl fun i hi => _ case h.e_a α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i S T : Finset ι hST : Disjoint S T hf_Indep : ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets i) hf_meas : ∀ (i : ι), Measurable (f i) πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πS : Set (Set Ω) := {s \| ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπS_pi : IsPiSystem πS hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s \| MeasurableSet s} πT : Set (Set Ω) := {s \| ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s} hπT_pi : IsPiSystem πT hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT s : Set ((i : { x // x ∈ S }) → β ↑i) sets_s : (i : { x // x ∈ S }) → Set (β ↑i) hs2 : Set.pi Set.univ sets_s = s t : Set ((i : { x // x ∈ T }) → β ↑i) sets_t : (i : { x // x ∈ T }) → Set (β ↑i) ht2 : Set.pi Set.univ sets_t = t hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i) ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i) sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi } h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i) h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i) h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) h_meas_inter : ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i) a : α h_indepS : ↑↑(↑κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(↑κ a) (f i ⁻¹' sets_s' i) h_indepT : ↑↑(↑κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(↑κ a) (f i ⁻¹' sets_t' i) h_indepST : ↑↑(↑κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) i : ι hi : i ∈ T ⊢ ↑↑(↑κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ↑↑(↑κ a) (f i ⁻¹' sets_t' i) rw [h_sets_s'_univ hi, Set.univ_inter] Qed
ProbabilityTheory.kernel.iIndepFun.indepFun_prod α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ classical have h_right : f k = (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘ fun a (j : ({k} : Finset ι)) => f j a := rfl have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _ let s : Finset ι := {i, j} have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘ fun a (j : s) => f j a := by ext1 a simp only [Prod.mk.inj_iff] constructor have h_meas_left : Measurable fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) := Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _) rw [h_left, h_right] refine' (hf_Indep.indepFun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right rw [Finset.disjoint_singleton_right] simp only [Finset.mem_insert, Finset.mem_singleton, not_or] exact ⟨hik.symm, hjk.symm⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_right : f k = (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘ fun a (j : ({k} : Finset ι)) => f j a := rfl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ let s : Finset ι := {i, j} α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘ fun a (j : s) => f j a := by ext1 a simp only [Prod.mk.inj_iff] constructor α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ have h_meas_left : Measurable fun p : ∀ l : s, β l => (p ⟨i, Finset.mem_insert_self i _⟩, p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) := Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _) α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ IndepFun (fun a => (f i a, f j a)) (f k) κ rw [h_left, h_right] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ IndepFun ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a) ((fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a) κ refine' (hf_Indep.indepFun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ Disjoint s {k} rw [Finset.disjoint_singleton_right] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ ¬k ∈ s simp only [Finset.mem_insert, Finset.mem_singleton, not_or] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} h_left : (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) }) ⊢ ¬k = i ∧ ¬k = j exact ⟨hik.symm, hjk.symm⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} ⊢ (fun ω => (f i ω, f j ω)) = (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a ext1 a case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} a : Ω ⊢ (f i a, f j a) = ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j => f (↑j) a) a simp only [Prod.mk.inj_iff] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝ : IsMarkovKernel κ ι : Type u_8 β : ι → Type u_9 m : (i : ι) → MeasurableSpace (β i) f : (i : ι) → Ω → β i hf_Indep : iIndepFun m f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k h_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) } s : Finset ι := {i, j} a : Ω ⊢ (f i a, f j a) = ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j_1 => f (↑j_1) a) a constructor ** Qed
ProbabilityTheory.kernel.iIndepFun.mul ** α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : Mul β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k ⊢ IndepFun (f i * f j) (f k) κ have : IndepFun (fun ω => (f i ω, f j ω)) (f k) κ μ := hf_Indep.indepFun_prod hf_meas i j k hik hjk α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : Mul β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k this : IndepFun (fun ω => (f i ω, f j ω)) (f k) κ ⊢ IndepFun (f i * f j) (f k) κ ** change IndepFun ((fun p : β × β => p.fst * p.snd) ∘ fun ω => (f i ω, f j ω)) (id ∘ f k) κ μ ** α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : Mul β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) i j k : ι hik : i ≠ k hjk : j ≠ k this : IndepFun (fun ω => (f i ω, f j ω)) (f k) κ ⊢ IndepFun ((fun p => p.1 * p.2) ∘ fun ω => (f i ω, f j ω)) (id ∘ f k) κ exact IndepFun.comp this (measurable_fst.mul measurable_snd) measurable_id Qed
ProbabilityTheory.kernel.iIndepFun.indepFun_finset_prod_of_not_mem α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s ⊢ IndepFun (∏ j in s, f j) (f i) κ classical have h_right : f i = (fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘ fun a (j : ({i} : Finset ι)) => f j a := rfl have h_meas_right : Measurable fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ := measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩ have h_left : ∏ j in s, f j = (fun p : ∀ _j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a := by ext1 a simp only [Function.comp_apply] have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply] rw [this, Finset.prod_coe_sort] have h_meas_left : Measurable fun p : ∀ _j : s, β => ∏ j, p j := Finset.univ.measurable_prod fun (j : ↥s) (_H : j ∈ Finset.univ) => measurable_pi_apply j rw [h_left, h_right] exact (hf_Indep.indepFun_finset s {i} (Finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp h_meas_left h_meas_right α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_right : f i = (fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩) ∘ fun a (j : ({i} : Finset ι)) => f j a := rfl α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_meas_right : Measurable fun p : ∀ _j : ({i} : Finset ι), β => p ⟨i, Finset.mem_singleton_self i⟩ := measurable_pi_apply ⟨i, Finset.mem_singleton_self i⟩ α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_left : ∏ j in s, f j = (fun p : ∀ _j : s, β => ∏ j, p j) ∘ fun a (j : s) => f j a := by ext1 a simp only [Function.comp_apply] have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply] rw [this, Finset.prod_coe_sort] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } h_left : ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a ⊢ IndepFun (∏ j in s, f j) (f i) κ have h_meas_left : Measurable fun p : ∀ _j : s, β => ∏ j, p j := Finset.univ.measurable_prod fun (j : ↥s) (_H : j ∈ Finset.univ) => measurable_pi_apply j α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } h_left : ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => ∏ j : { x // x ∈ s }, p j ⊢ IndepFun (∏ j in s, f j) (f i) κ rw [h_left, h_right] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } h_left : ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a h_meas_left : Measurable fun p => ∏ j : { x // x ∈ s }, p j ⊢ IndepFun ((fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a) ((fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a) κ exact (hf_Indep.indepFun_finset s {i} (Finset.disjoint_singleton_left.mpr hi).symm hf_meas).comp h_meas_left h_meas_right α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } ⊢ ∏ j in s, f j = (fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a ext1 a case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω ⊢ Finset.prod s (fun j => f j) a = ((fun p => ∏ j : { x // x ∈ s }, p j) ∘ fun a j => f (↑j) a) a simp only [Function.comp_apply] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω ⊢ Finset.prod s (fun j => f j) a = ∏ j : { x // x ∈ s }, f (↑j) a have : (∏ j : ↥s, f (↑j) a) = (∏ j : ↥s, f ↑j) a := by rw [Finset.prod_apply] case h α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω this : ∏ j : { x // x ∈ s }, f (↑j) a = Finset.prod Finset.univ (fun j => f ↑j) a ⊢ Finset.prod s (fun j => f j) a = ∏ j : { x // x ∈ s }, f (↑j) a rw [this, Finset.prod_coe_sort] α : Type u_1 Ω : Type u_2 ι✝ : Type u_3 β✝ : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f✝ : Ω → β✝ g : Ω → β' inst✝² : IsMarkovKernel κ ι : Type u_8 β : Type u_9 m : MeasurableSpace β inst✝¹ : CommMonoid β inst✝ : MeasurableMul₂ β f : ι → Ω → β hf_Indep : iIndepFun (fun x => m) f κ hf_meas : ∀ (i : ι), Measurable (f i) s : Finset ι i : ι hi : ¬i ∈ s h_right : f i = (fun p => p { val := i, property := (_ : i ∈ {i}) }) ∘ fun a j => f (↑j) a h_meas_right : Measurable fun p => p { val := i, property := (_ : i ∈ {i}) } a : Ω ⊢ ∏ j : { x // x ∈ s }, f (↑j) a = Finset.prod Finset.univ (fun j => f ↑j) a rw [Finset.prod_apply] ** Qed
ProbabilityTheory.kernel.iIndepSet.iIndepFun_indicator α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ ⊢ iIndepFun (fun _n => m) (fun n => Set.indicator (s n) fun _ω => 1) κ rw [iIndepFun_iff_measure_inter_preimage_eq_mul] α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ ⊢ ∀ (S : Finset ι) {sets : ι → Set β}, (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, (Set.indicator (s i) fun _ω => 1) ⁻¹' sets i) = ∏ i in S, ↑↑(↑κ a) ((Set.indicator (s i) fun _ω => 1) ⁻¹' sets i) rintro S π _hπ α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, (Set.indicator (s i) fun _ω => 1) ⁻¹' π i) = ∏ i in S, ↑↑(↑κ a) ((Set.indicator (s i) fun _ω => 1) ⁻¹' π i) simp_rw [Set.indicator_const_preimage_eq_union] α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) ⊢ ∀ᵐ (a : α) ∂μ, ↑↑(↑κ a) (⋂ i ∈ S, (if 1 ∈ π i then s i else ∅) ∪ if 0 ∈ π i then (s i)ᶜ else ∅) = ∏ x in S, ↑↑(↑κ a) ((if 1 ∈ π x then s x else ∅) ∪ if 0 ∈ π x then (s x)ᶜ else ∅) refine' @hs S (fun i => ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s i)ᶜ ∅) fun i _hi => _ α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S ⊢ (fun i => (if 1 ∈ π i then s i else ∅) ∪ if 0 ∈ π i then (s i)ᶜ else ∅) i ∈ (fun x => {s_1 \| MeasurableSet s_1}) i have hsi : MeasurableSet[generateFrom {s i}] (s i) := measurableSet_generateFrom (Set.mem_singleton _) α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S hsi : MeasurableSet (s i) ⊢ (fun i => (if 1 ∈ π i then s i else ∅) ∪ if 0 ∈ π i then (s i)ᶜ else ∅) i ∈ (fun x => {s_1 \| MeasurableSet s_1}) i refine' MeasurableSet.union (MeasurableSet.ite' (fun _ => hsi) fun _ => _) (MeasurableSet.ite' (fun _ => hsi.compl) fun _ => _) case refine'_1 α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S hsi : MeasurableSet (s i) x✝ : ¬1 ∈ π i ⊢ MeasurableSet ∅ exact @MeasurableSet.empty _ (generateFrom {s i}) case refine'_2 α : Type u_1 Ω : Type u_2 ι : Type u_3 β : Type u_4 β' : Type u_5 γ : Type u_6 γ' : Type u_7 _mα : MeasurableSpace α _mΩ : MeasurableSpace Ω κ : { x // x ∈ kernel α Ω } μ : Measure α f : Ω → β g : Ω → β' inst✝¹ : Zero β inst✝ : One β m : MeasurableSpace β s : ι → Set Ω hs : iIndepSet s κ S : Finset ι π : ι → Set β _hπ : ∀ (i : ι), i ∈ S → MeasurableSet (π i) i : ι _hi : i ∈ S hsi : MeasurableSet (s i) x✝ : ¬0 ∈ π i ⊢ MeasurableSet ∅ exact @MeasurableSet.empty _ (generateFrom {s i}) ** Qed
IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ⊢ Tendsto (fun j => closedBall (w j) (δ j)) l (VitaliFamily.filterAt (vitaliFamily μ K) x) refine' (vitaliFamily μ K).tendsto_filterAt_iff.mpr ⟨_, fun ε hε => _⟩ case refine'_1 α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ∈ VitaliFamily.setsAt (vitaliFamily μ K) x filter_upwards [xmem, δlim self_mem_nhdsWithin] with j hj h'j case h α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) j : ι hj : x ∈ closedBall (w j) (K * δ j) h'j : j ∈ δ ⁻¹' Ioi 0 ⊢ closedBall (w j) (δ j) ∈ VitaliFamily.setsAt (vitaliFamily μ K) x exact closedBall_mem_vitaliFamily_of_dist_le_mul μ hj h'j case refine'_2 α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε rcases l.eq_or_neBot with rfl \| h case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε ** have hK : 0 ≤ K := by rcases (xmem.and (δlim self_mem_nhdsWithin)).exists with ⟨j, hj, h'j⟩ have : 0 ≤ K * δ j := nonempty_closedBall.1 ⟨x, hj⟩ exact (mul_nonneg_iff_left_nonneg_of_pos (mem_Ioi.1 h'j)).1 this ** case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε have δpos := eventually_mem_of_tendsto_nhdsWithin δlim case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε replace δlim := tendsto_nhds_of_tendsto_nhdsWithin δlim case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε replace hK : 0 < K + 1 case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 ⊢ ∀ᶠ (i : ι) in l, closedBall (w i) (δ i) ⊆ closedBall x ε apply (((Metric.tendsto_nhds.mp δlim _ (div_pos hε hK)).and δpos).and xmem).mono case refine'_2.inr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 ⊢ ∀ (x_1 : ι), (dist (δ x_1) 0 < ε / (K + 1) ∧ δ x_1 ∈ Ioi 0) ∧ x ∈ closedBall (w x_1) (K * δ x_1) → closedBall (w x_1) (δ x_1) ⊆ closedBall x ε rintro j ⟨⟨hjε, hj₀ : 0 < δ j⟩, hx⟩ y hy case refine'_2.inr.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hx : x ∈ closedBall (w j) (K * δ j) hjε : dist (δ j) 0 < ε / (K + 1) hj₀ : 0 < δ j y : α hy : y ∈ closedBall (w j) (δ j) ⊢ y ∈ closedBall x ε ** replace hjε : (K + 1) * δ j < ε := by simpa [abs_eq_self.mpr hj₀.le] using (lt_div_iff' hK).mp hjε ** case refine'_2.inr.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hx : x ∈ closedBall (w j) (K * δ j) hj₀ : 0 < δ j y : α hy : y ∈ closedBall (w j) (δ j) hjε : (K + 1) * δ j < ε ⊢ y ∈ closedBall x ε simp only [mem_closedBall] at hx hy ⊢ case refine'_2.inr.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hj₀ : 0 < δ j y : α hjε : (K + 1) * δ j < ε hx : dist x (w j) ≤ K * δ j hy : dist y (w j) ≤ δ j ⊢ dist y x ≤ ε linarith [dist_triangle_right y x (w j)] case refine'_2.inl α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 w : ι → α δ : ι → ℝ ε : ℝ hε : ε > 0 δlim : Tendsto δ ⊥ (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in ⊥, x ∈ closedBall (w j) (K * δ j) ⊢ ∀ᶠ (i : ι) in ⊥, closedBall (w i) (δ i) ⊆ closedBall x ε simp α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l ⊢ 0 ≤ K rcases (xmem.and (δlim self_mem_nhdsWithin)).exists with ⟨j, hj, h'j⟩ case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l j : ι hj : x ∈ closedBall (w j) (K * δ j) h'j : δ j ∈ Ioi 0 ⊢ 0 ≤ K ** have : 0 ≤ K * δ j := nonempty_closedBall.1 ⟨x, hj⟩ ** case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ δlim : Tendsto δ l (𝓝[Ioi 0] 0) xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l j : ι hj : x ∈ closedBall (w j) (K * δ j) h'j : δ j ∈ Ioi 0 this : 0 ≤ K * δ j ⊢ 0 ≤ K exact (mul_nonneg_iff_left_nonneg_of_pos (mem_Ioi.1 h'j)).1 this case hK α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l hK : 0 ≤ K δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) ⊢ 0 < K + 1 linarith α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : MeasurableSpace α μ : Measure α inst✝³ : IsUnifLocDoublingMeasure μ inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α inst✝ : IsLocallyFiniteMeasure μ K : ℝ x : α ι : Type u_2 l : Filter ι w : ι → α δ : ι → ℝ xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j) ε : ℝ hε : ε > 0 h : NeBot l δpos : ∀ᶠ (i : ι) in l, δ i ∈ Ioi 0 δlim : Tendsto δ l (𝓝 0) hK : 0 < K + 1 j : ι hx : x ∈ closedBall (w j) (K * δ j) hjε : dist (δ j) 0 < ε / (K + 1) hj₀ : 0 < δ j y : α hy : y ∈ closedBall (w j) (δ j) ⊢ (K + 1) * δ j < ε simpa [abs_eq_self.mpr hj₀.le] using (lt_div_iff' hK).mp hjε Qed
IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div ** α : Type u_1 inst✝⁶ : MetricSpace α inst✝⁵ : MeasurableSpace α μ : Measure α inst✝⁴ : IsUnifLocDoublingMeasure μ inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ E : Type u_2 inst✝ : NormedAddCommGroup E S : Set α K : ℝ ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ S, ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[Ioi 0] 0) → (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) → Tendsto (fun j => ↑↑μ (S ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1) filter_upwards [(vitaliFamily μ K).ae_tendsto_measure_inter_div S] with x hx ι l w δ δlim xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem) Qed
IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub ** α : Type u_1 inst✝⁶ : MetricSpace α inst✝⁵ : MeasurableSpace α μ : Measure α inst✝⁴ : IsUnifLocDoublingMeasure μ inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ E : Type u_2 inst✝ : NormedAddCommGroup E f : α → E hf : LocallyIntegrable f K : ℝ ⊢ ∀ᵐ (x : α) ∂μ, ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[Ioi 0] 0) → (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) → Tendsto (fun j => ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0) filter_upwards [(vitaliFamily μ K).ae_tendsto_average_norm_sub hf] with x hx ι l w δ δlim xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem) Qed
MeasureTheory.measurable_measure_mul_right ** G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ Measurable fun x => ↑↑μ ((fun y => y * x) ⁻¹' s) ** suffices Measurable fun y => μ ((fun x => (x, y)) ⁻¹' ((fun z : G × G => ((1 : G), z.1 * z.2)) ⁻¹' univ ×ˢ s)) by convert this using 1; ext1 x; congr 1 with y : 1; simp ** G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) apply measurable_measure_prod_mk_right case hs G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ MeasurableSet ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s) apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs) G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s ⊢ MeasurableSpace G infer_instance G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) ⊢ Measurable fun x => ↑↑μ ((fun y => y * x) ⁻¹' s) convert this using 1 case h.e'_5 G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) ⊢ (fun x => ↑↑μ ((fun y => y * x) ⁻¹' s)) = fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) ext1 x case h.e'_5.h G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) x : G ⊢ ↑↑μ ((fun y => y * x) ⁻¹' s) = ↑↑μ ((fun x_1 => (x_1, x)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) congr 1 with y : 1 case h.e'_5.h.e_a.h G : Type u_1 inst✝⁴ : MeasurableSpace G inst✝³ : Group G inst✝² : MeasurableMul₂ G μ ν : Measure G inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set G hs : MeasurableSet s this : Measurable fun y => ↑↑μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s)) x y : G ⊢ y ∈ (fun y => y * x) ⁻¹' s ↔ y ∈ (fun x_1 => (x_1, x)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s) simp Qed
MeasureTheory.measure_inv_null G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ ⊢ ↑↑μ s⁻¹ = 0 ↔ ↑↑μ s = 0 refine' ⟨fun hs => _, (quasiMeasurePreserving_inv μ).preimage_null⟩ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ hs : ↑↑μ s⁻¹ = 0 ⊢ ↑↑μ s = 0 rw [← inv_inv s] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ hs : ↑↑μ s⁻¹ = 0 ⊢ ↑↑μ s⁻¹⁻¹ = 0 exact (quasiMeasurePreserving_inv μ).preimage_null hs ** Qed
MeasureTheory.absolutelyContinuous_inv G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ ⊢ μ ≪ Measure.inv μ refine' AbsolutelyContinuous.mk fun s _ => _ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ s : Set G x✝ : MeasurableSet s ⊢ ↑↑(Measure.inv μ) s = 0 → ↑↑μ s = 0 simp_rw [inv_apply μ s, measure_inv_null, imp_self] ** Qed
MeasureTheory.lintegral_lintegral_mul_inv ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) ⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ ** have h : Measurable fun z : G × G => (z.2 * z.1, z.1⁻¹) := (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) ⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ ** have h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν) := hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv μ ν).quasiMeasurePreserving ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ simp_rw [lintegral_lintegral h2f, lintegral_lintegral hf] G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂Measure.prod μ ν = ∫⁻ (z : G × G), f z.1 z.2 ∂Measure.prod μ ν conv_rhs => rw [← (measurePreserving_mul_prod_inv μ ν).map_eq] G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂Measure.prod μ ν = ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (Measure.prod μ ν) symm G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν f : G → G → ℝ≥0∞ hf : AEMeasurable (uncurry f) h : Measurable fun z => (z.2 * z.1, z.1⁻¹) h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) ⊢ ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (Measure.prod μ ν) = ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂Measure.prod μ ν exact lintegral_map' (hf.mono' (measurePreserving_mul_prod_inv μ ν).map_eq.absolutelyContinuous) h.aemeasurable Qed
MeasureTheory.measure_mul_right_null ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ y : G ⊢ ↑↑μ ((fun x => x * y) ⁻¹' s) = 0 ↔ ↑↑μ ((fun x => y⁻¹ * x) ⁻¹' s⁻¹)⁻¹ = 0 simp_rw [← inv_preimage, preimage_preimage, mul_inv_rev, inv_inv] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ y : G ⊢ ↑↑μ ((fun x => y⁻¹ * x) ⁻¹' s⁻¹)⁻¹ = 0 ↔ ↑↑μ s = 0 simp only [measure_inv_null μ, measure_preimage_mul] Qed
MeasureTheory.absolutelyContinuous_map_mul_right ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun x => x * g) μ refine' AbsolutelyContinuous.mk fun s hs => _ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑(map (fun x => x * g) μ) s = 0 → ↑↑μ s = 0 rw [map_apply (measurable_mul_const g) hs, measure_mul_right_null] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑μ s = 0 → ↑↑μ s = 0 exact id Qed
MeasureTheory.absolutelyContinuous_map_div_left G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun h => g / h) μ simp_rw [div_eq_mul_inv] ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun h => g * h⁻¹) μ erw [← map_map (measurable_const_mul g) measurable_inv] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ μ ≪ map (fun x => g * x) (map Inv.inv μ) conv_lhs => rw [← map_mul_left_eq_self μ g] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ map (fun x => g * x) μ ≪ map (fun x => g * x) (map Inv.inv μ) exact (absolutelyContinuous_inv μ).map (measurable_const_mul g) Qed
MeasureTheory.absolutelyContinuous_of_isMulLeftInvariant G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 ⊢ μ ≪ ν refine' AbsolutelyContinuous.mk fun s sm hνs => _ G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 s : Set G sm : MeasurableSet s hνs : ↑↑ν s = 0 ⊢ ↑↑μ s = 0 have h1 := measure_mul_lintegral_eq μ ν sm 1 measurable_one ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 s : Set G sm : MeasurableSet s hνs : ↑↑ν s = 0 h1 : ↑↑μ s * ∫⁻ (y : G), OfNat.ofNat 1 y ∂ν = ∫⁻ (x : G), ↑↑ν ((fun z => z * x) ⁻¹' s) * OfNat.ofNat 1 x⁻¹ ∂μ ⊢ ↑↑μ s = 0 simp_rw [Pi.one_apply, lintegral_one, mul_one, (measure_mul_right_null ν _).mpr hνs, lintegral_zero, mul_eq_zero (M₀ := ℝ≥0∞), measure_univ_eq_zero.not.mpr hν, or_false_iff] at h1 G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν hν : ν ≠ 0 s : Set G sm : MeasurableSet s hνs : ↑↑ν s = 0 h1 : ↑↑μ s = 0 ⊢ ↑↑μ s = 0 exact h1 Qed
MeasureTheory.measure_mul_measure_eq ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t have h1 := measure_lintegral_div_measure ν ν hs h2s h3s (t.indicator fun _ => 1) (measurable_const.indicator ht) G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ h1 : ↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t have h2 := measure_lintegral_div_measure μ ν hs h2s h3s (t.indicator fun _ => 1) (measurable_const.indicator ht) G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ h1 : ↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν h2 : ↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ∫⁻ (x : G), indicator t (fun x => 1) x ∂μ ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t rw [lintegral_indicator _ ht, set_lintegral_one] at h1 h2 G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s✝ : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulLeftInvariant μ inst✝ : IsMulLeftInvariant ν s t : Set G hs : MeasurableSet s ht : MeasurableSet t h2s : ↑↑ν s ≠ 0 h3s : ↑↑ν s ≠ ⊤ h1 : ↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ↑↑ν t h2 : ↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν = ↑↑μ t ⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t rw [← h1, mul_left_comm, h2] Qed
MeasureTheory.measurePreserving_mul_prod G : Type u_1 inst✝⁵ : MeasurableSpace G inst✝⁴ : Group G inst✝³ : MeasurableMul₂ G μ ν : Measure G inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ s : Set G inst✝ : IsMulRightInvariant μ ⊢ MeasurePreserving ?m.111270 apply measurePreserving_prod_mul_swap_right μ ν ** Qed
MeasureTheory.measurePreserving_div_prod G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulRightInvariant μ ⊢ MeasurePreserving ?m.118172 apply measurePreserving_prod_div_swap μ ν ** Qed
MeasureTheory.measurePreserving_mul_prod_inv_right ** G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulRightInvariant μ inst✝ : IsMulRightInvariant ν ⊢ MeasurePreserving fun z => (z.1 * z.2, z.1⁻¹) convert (measurePreserving_prod_div_swap ν μ).comp (measurePreserving_prod_mul_swap_right μ ν) using 1 case h.e'_5 G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulRightInvariant μ inst✝ : IsMulRightInvariant ν ⊢ (fun z => (z.1 * z.2, z.1⁻¹)) = (fun z => (z.2, z.1 / z.2)) ∘ fun z => (z.2, z.1 * z.2) ext1 ⟨x, y⟩ case h.e'_5.h.mk G : Type u_1 inst✝⁷ : MeasurableSpace G inst✝⁶ : Group G inst✝⁵ : MeasurableMul₂ G μ ν : Measure G inst✝⁴ : SigmaFinite ν inst✝³ : SigmaFinite μ s : Set G inst✝² : MeasurableInv G inst✝¹ : IsMulRightInvariant μ inst✝ : IsMulRightInvariant ν x y : G ⊢ ((x, y).1 * (x, y).2, (x, y).1⁻¹) = ((fun z => (z.2, z.1 / z.2)) ∘ fun z => (z.2, z.1 * z.2)) (x, y) simp_rw [Function.comp_apply, div_mul_eq_div_div_swap, div_self', one_div] Qed
MeasureTheory.quasiMeasurePreserving_inv_of_right_invariant G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulRightInvariant μ ⊢ QuasiMeasurePreserving Inv.inv exact (quasiMeasurePreserving_inv μ.inv).mono (inv_absolutelyContinuous μ.inv) (absolutelyContinuous_inv μ.inv) ** Qed
MeasureTheory.quasiMeasurePreserving_div_left G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ QuasiMeasurePreserving fun h => g / h simp_rw [div_eq_mul_inv] ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ QuasiMeasurePreserving fun h => g * h⁻¹ exact (measurePreserving_mul_left μ g).quasiMeasurePreserving.comp (quasiMeasurePreserving_inv μ) Qed
MeasureTheory.quasiMeasurePreserving_mul_right ** G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G ⊢ QuasiMeasurePreserving fun h => h * g refine' ⟨measurable_mul_const g, AbsolutelyContinuous.mk fun s hs => _⟩ G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑μ s = 0 → ↑↑(map (fun h => h * g) μ) s = 0 rw [map_apply (measurable_mul_const g) hs, measure_mul_right_null] G : Type u_1 inst✝⁶ : MeasurableSpace G inst✝⁵ : Group G inst✝⁴ : MeasurableMul₂ G μ ν : Measure G inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ s✝ : Set G inst✝¹ : MeasurableInv G inst✝ : IsMulLeftInvariant μ g : G s : Set G hs : MeasurableSet s ⊢ ↑↑μ s = 0 → ↑↑μ s = 0 exact id Qed
MeasureTheory.SignedMeasure.findExistsOneDivLT_min α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i m : ℕ hm : m < MeasureTheory.SignedMeasure.findExistsOneDivLT s i ⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m rw [findExistsOneDivLT, dif_pos hi] at hm α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i m : ℕ hm : m < Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n) ⊢ ¬MeasureTheory.SignedMeasure.ExistsOneDivLT s i m exact Nat.find_min _ hm ** Qed
MeasureTheory.SignedMeasure.someExistsOneDivLT_subset α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i by_cases hi : ¬s ≤[i] 0 case pos α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i ⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i exact let ⟨h, _⟩ := someExistsOneDivLT_spec hi h case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s i ⊆ i rw [someExistsOneDivLT, dif_neg hi] case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ ∅ ⊆ i exact Set.empty_subset _ ** Qed
MeasureTheory.SignedMeasure.someExistsOneDivLT_measurableSet α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) by_cases hi : ¬s ≤[i] 0 case pos α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬restrict s i ≤ restrict 0 i ⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) exact let ⟨_, h, _⟩ := someExistsOneDivLT_spec hi h case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s i) rw [someExistsOneDivLT, dif_neg hi] case neg α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi : ¬¬restrict s i ≤ restrict 0 i ⊢ MeasurableSet ∅ exact MeasurableSet.empty ** Qed
MeasureTheory.SignedMeasure.restrictNonposSeq_succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n) = MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) rw [restrictNonposSeq] ** Qed
MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n) cases n with \| zero => rw [restrictNonposSeq]; rw [← @Set.diff_empty _ i] at hi₂ rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩ exact lt_trans Nat.one_div_pos_of_nat h \| succ n => rw [restrictNonposSeq_succ] have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn convert measurable_of_not_restrict_le_zero _ hn using 3 exact funext fun x => by rw [Nat.lt_succ_iff] rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩ exact lt_trans Nat.one_div_pos_of_nat h case zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero) rw [restrictNonposSeq] case zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) rw [← @Set.diff_empty _ i] at hi₂ case zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅) hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) rcases someExistsOneDivLT_spec hi₂ with ⟨_, _, h⟩ case zero.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s (i \ ∅) ≤ restrict 0 (i \ ∅) hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.zero), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) left✝¹ : MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅) ⊆ i \ ∅ left✝ : MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) h : 1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ∅)) + 1) < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ∅)) exact lt_trans Nat.one_div_pos_of_nat h case succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n)) rw [restrictNonposSeq_succ] case succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) have h₁ : ¬s ≤[i \ ⋃ (k : ℕ) (_ : k ≤ n), restrictNonposSeq s i k] 0 := by refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn convert measurable_of_not_restrict_le_zero _ hn using 3 exact funext fun x => by rw [Nat.lt_succ_iff] case succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) h₁ : ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩ case succ.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) h₁ : ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) left✝¹ : MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊆ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k left✝ : MeasurableSet (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) h : 1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) + 1) < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) ⊢ 0 < ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) exact lt_trans Nat.one_div_pos_of_nat h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ ¬restrict s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) refine' mt (restrict_le_zero_subset _ _ (by simp [Nat.lt_succ_iff]; rfl)) hn α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ MeasurableSet (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) convert measurable_of_not_restrict_le_zero _ hn using 3 case h.e'_3.h.e'_4.h.e'_3 α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ (fun k => ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) = fun k => ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k exact funext fun x => by rw [Nat.lt_succ_iff] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k simp [Nat.lt_succ_iff] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k ⊆ i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k rfl α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α hi₂ : ¬restrict s i ≤ restrict 0 i n : ℕ hn : ¬restrict s (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤ restrict 0 (i \ ⋃ k, ⋃ (_ : k < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) x : ℕ ⊢ ⋃ (_ : x ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x = ⋃ (_ : x < Nat.succ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i x rw [Nat.lt_succ_iff] ** Qed
MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n < m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rw [Set.eq_empty_iff_forall_not_mem] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n < m ⊢ ∀ (x : α), ¬x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m rintro x ⟨hx₁, hx₂⟩ case intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n < m x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i m ⊢ False cases m case intro.zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n h : n < Nat.zero hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero ⊢ False rw [Nat.zero_eq] at h case intro.zero α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n h : n < 0 hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i Nat.zero ⊢ False linarith case intro.succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n n✝ : ℕ h : n < Nat.succ n✝ hx₂ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝) ⊢ False rw [restrictNonposSeq] at hx₂ case intro.succ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n : ℕ x : α hx₁ : x ∈ MeasureTheory.SignedMeasure.restrictNonposSeq s i n n✝ : ℕ h : n < Nat.succ n✝ hx₂ : x ∈ MeasureTheory.SignedMeasure.someExistsOneDivLT s (i \ ⋃ k, ⋃ (H : k ≤ n✝), let_fun this := (_ : k < Nat.succ n✝); MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊢ False exact (someExistsOneDivLT_subset hx₂).2 (Set.mem_iUnion.2 ⟨n, Set.mem_iUnion.2 ⟨Nat.lt_succ_iff.mp h, hx₁⟩⟩) ** Qed
MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ Pairwise (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i) intro n m h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n ≠ m ⊢ (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i) n m rw [Function.onFun, Set.disjoint_iff_inter_eq_empty] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h : n ≠ m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rcases lt_or_gt_of_ne h with (h \| h) case inl α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h✝ : n ≠ m h : n < m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rw [restrictNonposSeq_disjoint' h] case inr α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α n m : ℕ h✝ : n ≠ m h : n > m ⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅ rw [Set.inter_comm, restrictNonposSeq_disjoint' h] ** Qed
MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ BddBelow (measureOfNegatives s) simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α ⊢ ∃ x, ∀ (x_1 : ℝ), x_1 ∈ measureOfNegatives s → x ≤ x_1 by_contra' h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x ⊢ False have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x h' : ∀ (n : ℕ), ∃ y, y ∈ measureOfNegatives s ∧ y < -↑n ⊢ False choose f hf using h' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n ⊢ False have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by intro n rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩ exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n hf' : ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n ⊢ False choose B hmeas hr h_lt using hf' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n ⊢ False set A := ⋃ n, B n with hA α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n hfalse : ∀ (n : ℕ), ↑s A ≤ -↑n ⊢ False rcases exists_nat_gt (-s A) with ⟨n, hn⟩ case intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n hfalse : ∀ (n : ℕ), ↑s A ≤ -↑n n : ℕ hn : -↑s A < ↑n ⊢ False exact lt_irrefl _ ((neg_lt.1 hn).trans_le (hfalse n)) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n ⊢ ∀ (n : ℕ), ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n intro n α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n n : ℕ ⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩ case intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n n : ℕ hlt : f n < -↑n B : Set α hB₂ : ↑s B = f n hB₁ : MeasurableSet B hBr : restrict s B ≤ restrict 0 B ⊢ ∃ B, MeasurableSet B ∧ restrict s B ≤ restrict 0 B ∧ ↑s B < -↑n exact ⟨B, hB₁, hBr, hB₂.symm ▸ hlt⟩ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n ⊢ ∀ (n : ℕ), ↑s A ≤ -↑n intro n α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s A ≤ -↑n refine' le_trans _ (le_of_lt (h_lt _)) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s A ≤ ↑s (B n) rw [hA, ← Set.diff_union_of_subset (Set.subset_iUnion _ n), of_union Set.disjoint_sdiff_left _ (hmeas n)] α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s ((⋃ i, B i) \ B n) + ↑s (B n) ≤ ↑s (B n) refine' add_le_of_nonpos_left _ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0 have : s ≤[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ this : restrict s A ≤ restrict 0 A ⊢ ↑s ((⋃ i, B i) \ B n) ≤ 0 refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this) α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ this : restrict s A ≤ restrict 0 A ⊢ MeasurableSet (⋃ i, B i) exact MeasurableSet.iUnion hmeas α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s : SignedMeasure α i j : Set α h : ∀ (x : ℝ), ∃ x_1, x_1 ∈ measureOfNegatives s ∧ x_1 < x f : ℕ → ℝ hf : ∀ (n : ℕ), f n ∈ measureOfNegatives s ∧ f n < -↑n B : ℕ → Set α hmeas : ∀ (n : ℕ), MeasurableSet (B n) hr : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n) h_lt : ∀ (n : ℕ), ↑s (B n) < -↑n A : Set α := ⋃ n, B n hA : A = ⋃ n, B n n : ℕ ⊢ MeasurableSet ((⋃ i, B i) \ B n) exact (MeasurableSet.iUnion hmeas).diff (hmeas n) ** Qed
MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0 rw [restrict_le_restrict_iff s 0, restrict_le_restrict_iff 0 s] at hi' hj' α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ ↑s (i ∆ j) = 0 ∧ ↑s (iᶜ ∆ jᶜ) = 0 α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet j α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet i α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet jᶜ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet iᶜ constructor α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet j α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet i α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet jᶜ α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet iᶜ all_goals measurability case left α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ ↑s (i ∆ j) = 0 rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, of_union, le_antisymm (hi'.2 (hi.compl.inter hj) (Set.inter_subset_left _ _)) (hj'.1 (hi.compl.inter hj) (Set.inter_subset_right _ _)), le_antisymm (hj'.2 (hj.compl.inter hi) (Set.inter_subset_left _ _)) (hi'.1 (hj.compl.inter hi) (Set.inter_subset_right _ _)), zero_apply, zero_apply, zero_add] case left.h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ Disjoint (jᶜ ∩ i) (iᶜ ∩ j) exact Set.disjoint_of_subset_left (Set.inter_subset_left _ _) (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) (disjoint_comm.1 (IsCompl.disjoint isCompl_compl))) case left.hA α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (jᶜ ∩ i) exact hj.compl.inter hi case left.hB α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (iᶜ ∩ j) exact hi.compl.inter hj case right α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ ↑s (iᶜ ∆ jᶜ) = 0 rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, compl_compl, compl_compl, of_union, le_antisymm (hi'.2 (hj.inter hi.compl) (Set.inter_subset_right _ _)) (hj'.1 (hj.inter hi.compl) (Set.inter_subset_left _ _)), le_antisymm (hj'.2 (hi.inter hj.compl) (Set.inter_subset_right _ _)) (hi'.1 (hi.inter hj.compl) (Set.inter_subset_left _ _)), zero_apply, zero_apply, zero_add] case right.h α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ Disjoint (j ∩ iᶜ) (i ∩ jᶜ) exact Set.disjoint_of_subset_left (Set.inter_subset_left _ _) (Set.disjoint_of_subset_right (Set.inter_subset_right _ _) (IsCompl.disjoint isCompl_compl)) case right.hA α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (j ∩ iᶜ) exact hj.inter hi.compl case right.hB α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑0 j ≤ ↑s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → ↑s j ≤ ↑0 j hj' : (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → ↑0 j_1 ≤ ↑s j_1) ∧ ∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ jᶜ → ↑s j_1 ≤ ↑0 j_1 ⊢ MeasurableSet (i ∩ jᶜ) exact hi.inter hj.compl α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α M : Type u_3 inst✝² : AddCommMonoid M inst✝¹ : TopologicalSpace M inst✝ : OrderedAddCommMonoid M s✝ : SignedMeasure α i✝ j✝ : Set α s : SignedMeasure α i j : Set α hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ hj' : restrict 0 j ≤ restrict s j ∧ restrict s jᶜ ≤ restrict 0 jᶜ ⊢ MeasurableSet iᶜ measurability ** Qed
MeasureTheory.hahn_decomposition α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let d : Set α → ℝ := fun s => ((μ s).toNNReal : ℝ) - (ν s).toNNReal α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let c : Set ℝ := d '' { s \| MeasurableSet s } α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let γ : ℝ := sSup c α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hμ : ∀ s, μ s ≠ ∞ := measure_ne_top μ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hν : ∀ s, ν s ≠ ∞ := measure_ne_top ν α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have to_nnreal_μ : ∀ s, ((μ s).toNNReal : ℝ≥0∞) = μ s := fun s => ENNReal.coe_toNNReal <\| hμ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have to_nnreal_ν : ∀ s, ((ν s).toNNReal : ℝ≥0∞) = ν s := fun s => ENNReal.coe_toNNReal <\| hν _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_split : ∀ s t, MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) := by intro s t _hs ht dsimp only rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht, ENNReal.toNNReal_add (hμ _) (hμ _), ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add, NNReal.coe_add] simp only [sub_eq_add_neg, neg_add] abel α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_Union : ∀ s : ℕ → Set α, Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) := by intro s hm refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal _).comp <\| tendsto_measure_iUnion hm exact hμ _ exact hν _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_Inter : ∀ s : ℕ → Set α, (∀ n, MeasurableSet (s n)) → (∀ n m, n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) := by intro s hs hm refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal <\| _).comp <\| tendsto_measure_iInter hs hm _ exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have bdd_c : BddAbove c := by use (μ univ).toNNReal rintro r ⟨s, _hs, rfl⟩ refine' le_trans (sub_le_self _ <\| NNReal.coe_nonneg _) _ rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ] exact measure_mono (subset_univ _) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have c_nonempty : c.Nonempty := Nonempty.image _ ⟨_, MeasurableSet.empty⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have d_le_γ : ∀ s, MeasurableSet s → d s ≤ γ := fun s hs => le_csSup bdd_c ⟨s, hs, rfl⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have : ∀ n : ℕ, ∃ s : Set α, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s := by intro n have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n) rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩ exact ⟨s, hs, hlt⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t rcases Classical.axiom_of_choice this with ⟨e, he⟩ case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have he₁ : ∀ n, MeasurableSet (e n) := fun n => (he n).1 case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have he₂ : ∀ n, γ - (1 / 2) ^ n < d (e n) := fun n => (he n).2 case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let f : ℕ → ℕ → Set α := fun n m => (Finset.Ico n (m + 1)).inf e case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hf : ∀ n m, MeasurableSet (f n m) := by intro n m simp only [Finset.inf_eq_iInf] exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _ case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have f_subset_f : ∀ {a b c d}, a ≤ b → c ≤ d → f a d ⊆ f b c := by intro a b c d hab hcd simp_rw [Finset.inf_eq_iInf] exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <\| Nat.succ_le_succ hcd) case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have f_succ : ∀ n m, n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) := by intro n m hnm have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm) simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm] rfl ** case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t let s := ⋃ m, ⋂ n, f m n case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t have hs : MeasurableSet s := MeasurableSet.iUnion fun n => MeasurableSet.iInter fun m => hf _ _ case intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s ⊢ ∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t refine' ⟨s, hs, _, _⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s ⊢ ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) intro s t _hs ht α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ d s = d (s \ t) + d (s ∩ t) dsimp only α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) = ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \ t))) + (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t)))) rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht, ENNReal.toNNReal_add (hμ _) (hμ _), ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add, NNReal.coe_add] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - (↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑ν (s \ t)))) = ↑(ENNReal.toNNReal (↑↑μ (s \ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \ t))) + (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t)))) simp only [sub_eq_add_neg, neg_add] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s s t : Set α _hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \ t))) + (-↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t)))) = ↑(ENNReal.toNNReal (↑↑μ (s \ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \ t))) + (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s ∩ t)))) abel α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) ⊢ ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) intro s hm α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal _).comp <\| tendsto_measure_iUnion hm case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ ↑↑μ (⋃ n, s n) ≠ ⊤ case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ ↑↑ν (⋃ n, s n) ≠ ⊤ exact hμ _ case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) s : ℕ → Set α hm : Monotone s ⊢ ↑↑ν (⋃ n, s n) ≠ ⊤ exact hν _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) ⊢ ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) intro s hs hm α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) refine' Tendsto.sub _ _ <;> refine' NNReal.tendsto_coe.2 <\| (ENNReal.tendsto_toNNReal <\| _).comp <\| tendsto_measure_iInter hs hm _ case refine'_1.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ↑↑μ (⋂ n, s n) ≠ ⊤ case refine'_1.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ∃ i, ↑↑μ (s i) ≠ ⊤ case refine'_2.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ↑↑ν (⋂ n, s n) ≠ ⊤ case refine'_2.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) s : ℕ → Set α hs : ∀ (n : ℕ), MeasurableSet (s n) hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n ⊢ ∃ i, ↑↑ν (s i) ≠ ⊤ exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) ⊢ BddAbove c use (μ univ).toNNReal case h α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) ⊢ ↑(ENNReal.toNNReal (↑↑μ univ)) ∈ upperBounds c rintro r ⟨s, _hs, rfl⟩ case h.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) s : Set α _hs : s ∈ {s \| MeasurableSet s} ⊢ d s ≤ ↑(ENNReal.toNNReal (↑↑μ univ)) refine' le_trans (sub_le_self _ <\| NNReal.coe_nonneg _) _ case h.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) s : Set α _hs : s ∈ {s \| MeasurableSet s} ⊢ ↑(ENNReal.toNNReal (↑↑μ s)) ≤ ↑(ENNReal.toNNReal (↑↑μ univ)) rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ] case h.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) s : Set α _hs : s ∈ {s \| MeasurableSet s} ⊢ ↑↑μ s ≤ ↑↑μ univ exact measure_mono (subset_univ _) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ ⊢ ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s intro n α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ n : ℕ ⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ n : ℕ this : γ - (1 / 2) ^ n < γ ⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩ case intro.intro.intro.intro α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ n : ℕ this : γ - (1 / 2) ^ n < γ s : Set α hs : s ∈ {s \| MeasurableSet s} hlt : γ - (1 / 2) ^ n < d s ⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s exact ⟨s, hs, hlt⟩ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e ⊢ ∀ (n m : ℕ), MeasurableSet (f n m) intro n m α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e n m : ℕ ⊢ MeasurableSet (f n m) simp only [Finset.inf_eq_iInf] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e n m : ℕ ⊢ MeasurableSet (⨅ a ∈ Finset.Ico n (m + 1), e a) exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) ⊢ ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c intro a b c d hab hcd α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c✝ : Set ℝ := d✝ '' {s \| MeasurableSet s} γ : ℝ := sSup c✝ hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \ t) + d✝ (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ n, s n))) bdd_c : BddAbove c✝ c_nonempty : Set.Nonempty c✝ d_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) a b c d : ℕ hab : a ≤ b hcd : c ≤ d ⊢ f a d ⊆ f b c simp_rw [Finset.inf_eq_iInf] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c✝ : Set ℝ := d✝ '' {s \| MeasurableSet s} γ : ℝ := sSup c✝ hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \ t) + d✝ (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ n, s n))) bdd_c : BddAbove c✝ c_nonempty : Set.Nonempty c✝ d_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) a b c d : ℕ hab : a ≤ b hcd : c ≤ d ⊢ ⨅ a_1 ∈ Finset.Ico a (d + 1), e a_1 ⊆ ⨅ a ∈ Finset.Ico b (c + 1), e a exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <\| Nat.succ_le_succ hcd) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c ⊢ ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) intro n m hnm α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c n m : ℕ hnm : n ≤ m ⊢ f n (m + 1) = f n m ∩ e (m + 1) have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c n m : ℕ hnm : n ≤ m this : n ≤ m + 1 ⊢ f n (m + 1) = f n m ∩ e (m + 1) simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c n m : ℕ hnm : n ≤ m this : n ≤ m + 1 ⊢ e (m + 1) ⊓ Finset.inf (Finset.Ico n (m + 1)) e = e (m + 1) ∩ Finset.inf (Finset.Ico n (m + 1)) e rfl α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) ⊢ ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) intro n m h α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) refine' Nat.le_induction _ _ n h case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m) have := he₂ m case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n this : γ - (1 / 2) ^ m < d (e m) ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m) simp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton] case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n this : γ - (1 / 2) ^ m < d (e m) ⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) - ↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ ↑(ENNReal.toNNReal (↑↑μ (e m))) - ↑(ENNReal.toNNReal (↑↑ν (e m))) linarith case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n m : ℕ h : m ≤ n ⊢ ∀ (n : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1)) intro n (hmn : m ≤ n) ih case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1)) ** have : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) := by calc γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by refine' add_le_add_left (add_le_add_left _ _) γ simp only [pow_add, pow_one, le_sub_iff_add_le] linarith _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <\| he₂ _) ih) _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc] _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) := by rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left] abel exact (he₁ _).union (hf _ _) exact he₁ _ _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <\| (he₁ _).union (hf _ _)) _ ** case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) this : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) ⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1)) exact (add_le_add_iff_left γ).1 this α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) ** calc γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by refine' add_le_add_left (add_le_add_left _ _) γ simp only [pow_add, pow_one, le_sub_iff_add_le] linarith _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by simp only [sub_eq_add_neg]; abel _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <\| he₂ _) ih) _ ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) := by rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc] _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) := by rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left] abel exact (he₁ _).union (hf _ _) exact he₁ _ _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <\| (he₁ _).union (hf _ _)) _ ** α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) refine' add_le_add_left (add_le_add_left _ _) γ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ (1 / 2) ^ (n + 1) ≤ (1 / 2) ^ n - (1 / 2) ^ (n + 1) simp only [pow_add, pow_one, le_sub_iff_add_le] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ (1 / 2) ^ n * (1 / 2) + (1 / 2) ^ n * (1 / 2) ≤ (1 / 2) ^ n linarith α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) simp only [sub_eq_add_neg] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + -(2 * (1 / 2) ^ m) + ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) = sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + -(1 / 2) ^ (n + 1) + (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s \| MeasurableSet s}) + -(2 * (1 / 2) ^ m) + (1 / 2) ^ n) abel α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ d (e (n + 1)) + d (f m n) ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1)) = d (f m n \ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1)) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1) ∪ f m n) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1)) abel case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1) ∪ f m n) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1)) exact (he₁ _).union (hf _ _) case a α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) n✝ m : ℕ h : m ≤ n✝ n : ℕ hmn : m ≤ n ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) ⊢ MeasurableSet (e (n + 1)) exact he₁ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n ⊢ γ ≤ d s ** have hγ : Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) := by suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by simpa only [mul_zero, tsub_zero] exact tendsto_const_nhds.sub <\| tendsto_const_nhds.mul <\| tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <\| half_pos <\| zero_lt_one) (half_lt_self zero_lt_one) ** α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ⊢ γ ≤ d s have hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) := by refine' d_Union _ _ exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <\| f_subset_f hnm <\| le_rfl α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) ⊢ γ ≤ d s refine' le_of_tendsto_of_tendsto' hγ hd fun m => _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) ⊢ γ - 2 * (1 / 2) ^ m ≤ d (⋂ n, f m n) refine' ge_of_tendsto this (eventually_atTop.2 ⟨m, fun n hmn => _⟩) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) n : ℕ hmn : n ≥ m ⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n) refine' le_trans _ (le_d_f _ _ hmn) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ this : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) n : ℕ hmn : n ≥ m ⊢ γ - 2 * (1 / 2) ^ m ≤ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n exact le_add_of_le_of_nonneg le_rfl (pow_nonneg (le_of_lt <\| half_pos <\| zero_lt_one) _) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n ⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ** suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by simpa only [mul_zero, tsub_zero] ** α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n ⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) exact tendsto_const_nhds.sub <\| tendsto_const_nhds.mul <\| tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <\| half_pos <\| zero_lt_one) (half_lt_self zero_lt_one) α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n this : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) ⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) simpa only [mul_zero, tsub_zero] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ⊢ Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) refine' d_Union _ _ α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) ⊢ Monotone fun m => ⋂ n, f m n exact fun n m hnm => subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <\| f_subset_f hnm <\| le_rfl α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ ⊢ Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ n, f m n))) refine' d_Inter _ _ _ case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ ⊢ ∀ (n : ℕ), MeasurableSet (f m n) intro n case refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m n : ℕ ⊢ MeasurableSet (f m n) exact hf _ _ case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m : ℕ ⊢ ∀ (n m_1 : ℕ), n ≤ m_1 → f m m_1 ⊆ f m n intro n m hnm case refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n hγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) m✝ n m : ℕ hnm : n ≤ m ⊢ f m✝ m ⊆ f m✝ n exact f_subset_f le_rfl hnm case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s ⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t intro t ht hts case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ ↑↑ν t ≤ ↑↑μ t have : 0 ≤ d t := (add_le_add_iff_left γ).1 <\| calc γ + 0 ≤ d s := by rw [add_zero]; exact γ_le_d_s _ = d (s \ t) + d t := by rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts] _ ≤ γ + d t := add_le_add (d_le_γ _ (hs.diff ht)) le_rfl case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s this : 0 ≤ d t ⊢ ↑↑ν t ≤ ↑↑μ t rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe] case intro.refine'_1 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s this : 0 ≤ d t ⊢ ↑(ENNReal.toNNReal (↑↑ν t)) ≤ ↑(ENNReal.toNNReal (↑↑μ t)) simpa only [le_sub_iff_add_le, zero_add] using this α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ γ + 0 ≤ d s rw [add_zero] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ γ ≤ d s exact γ_le_d_s α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ s ⊢ d s = d (s \ t) + d t rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts] case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s ⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t intro t ht hts case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ ↑↑μ t ≤ ↑↑ν t have : d t ≤ 0 := (add_le_add_iff_left γ).1 <\| calc γ + d t ≤ d s + d t := add_le_add γ_le_d_s le_rfl _ = d (s ∪ t) := by rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right, (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left] _ ≤ γ + 0 := by rw [add_zero]; exact d_le_γ _ (hs.union ht) case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ this : d t ≤ 0 ⊢ ↑↑μ t ≤ ↑↑ν t rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe] case intro.refine'_2 α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ this : d t ≤ 0 ⊢ ↑(ENNReal.toNNReal (↑↑μ t)) ≤ ↑(ENNReal.toNNReal (↑↑ν t)) simpa only [sub_le_iff_le_add, zero_add] using this α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ d s + d t = d (s ∪ t) rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right, (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ d (s ∪ t) ≤ γ + 0 rw [add_zero] α : Type u_1 inst✝² : MeasurableSpace α μ ν : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : IsFiniteMeasure ν d : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) c : Set ℝ := d '' {s \| MeasurableSet s} γ : ℝ := sSup c hμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤ hν : ∀ (s : Set α), ↑↑ν s ≠ ⊤ to_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s to_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s d_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \ t) + d (s ∩ t) d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) bdd_c : BddAbove c c_nonempty : Set.Nonempty c d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s e : ℕ → Set α he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x) he₁ : ∀ (n : ℕ), MeasurableSet (e n) he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n) f : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e hf : ∀ (n m : ℕ), MeasurableSet (f n m) f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) s : Set α := ⋃ m, ⋂ n, f m n γ_le_d_s : γ ≤ d s hs : MeasurableSet s t : Set α ht : MeasurableSet t hts : t ⊆ sᶜ ⊢ d (s ∪ t) ≤ γ exact d_le_γ _ (hs.union ht) Qed
measurable_div_const' G : Type u_1 inst✝² : DivInvMonoid G inst✝¹ : MeasurableSpace G inst✝ : MeasurableMul G g : G ⊢ Measurable fun h => h / g simp_rw [div_eq_mul_inv, measurable_mul_const] ** Qed
nullMeasurableSet_eq_fun G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g ⊢ NullMeasurableSet {x \| f x = g x} apply (measurableSet_eq_fun hf.measurable_mk hg.measurable_mk).nullMeasurableSet.congr G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g ⊢ {x \| AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} =ᵐ[μ] {x \| f x = g x} filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx case h G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g x : α hfx : f x = AEMeasurable.mk f hf x hgx : g x = AEMeasurable.mk g hg x ⊢ setOf (fun x => AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) x = setOf (fun x => f x = g x) x change (hf.mk f x = hg.mk g x) = (f x = g x) case h G : Type u_1 α : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m : MeasurableSpace α f✝ g✝ : α → G μ : Measure α E : Type u_3 inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E f g : α → E hf : AEMeasurable f hg : AEMeasurable g x : α hfx : f x = AEMeasurable.mk f hf x hgx : g x = AEMeasurable.mk g hg x ⊢ (AEMeasurable.mk f hf x = AEMeasurable.mk g hg x) = (f x = g x) simp only [hfx, hgx] ** Qed
measurableSet_eq_fun_of_countable G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g ⊢ MeasurableSet {x \| f x = g x} have : { x \| f x = g x } = ⋃ j, { x \| f x = j } ∩ { x \| g x = j } := by ext1 x simp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right'] G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} ⊢ MeasurableSet {x \| f x = g x} rw [this] G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} ⊢ MeasurableSet (⋃ j, {x \| f x = j} ∩ {x \| g x = j}) refine' MeasurableSet.iUnion fun j => MeasurableSet.inter _ _ G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g ⊢ {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} ext1 x case h G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g x : α ⊢ x ∈ {x \| f x = g x} ↔ x ∈ ⋃ j, {x \| f x = j} ∩ {x \| g x = j} simp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right'] case refine'_1 G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} j : E ⊢ MeasurableSet {x \| f x = j} exact hf (measurableSet_singleton j) case refine'_2 G : Type u_1 α : Type u_2 inst✝⁴ : MeasurableSpace G inst✝³ : Div G m✝ : MeasurableSpace α f✝ g✝ : α → G μ : Measure α m : MeasurableSpace α E : Type u_3 inst✝² : MeasurableSpace E inst✝¹ : MeasurableSingletonClass E inst✝ : Countable E f g : α → E hf : Measurable f hg : Measurable g this : {x \| f x = g x} = ⋃ j, {x \| f x = j} ∩ {x \| g x = j} j : E ⊢ MeasurableSet {x \| g x = j} exact hg (measurableSet_singleton j) ** Qed
ae_eq_trim_of_measurable G : Type u_1 α✝ : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m✝ : MeasurableSpace α✝ f✝ g✝ : α✝ → G μ✝ : Measure α✝ α : Type u_3 E : Type u_4 m m0 : MeasurableSpace α μ : Measure α inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E hm : m ≤ m0 f g : α → E hf : Measurable f hg : Measurable g hfg : f =ᵐ[μ] g ⊢ f =ᵐ[Measure.trim μ hm] g rwa [Filter.EventuallyEq, @ae_iff _ m, trim_measurableSet_eq hm _] G : Type u_1 α✝ : Type u_2 inst✝⁵ : MeasurableSpace G inst✝⁴ : Div G m✝ : MeasurableSpace α✝ f✝ g✝ : α✝ → G μ✝ : Measure α✝ α : Type u_3 E : Type u_4 m m0 : MeasurableSpace α μ : Measure α inst✝³ : MeasurableSpace E inst✝² : AddGroup E inst✝¹ : MeasurableSingletonClass E inst✝ : MeasurableSub₂ E hm : m ≤ m0 f g : α → E hf : Measurable f hg : Measurable g hfg : f =ᵐ[μ] g ⊢ MeasurableSet {a \| ¬f a = g a} exact @MeasurableSet.compl α _ m (@measurableSet_eq_fun α m E _ _ _ _ _ _ hf hg) ** Qed
measurable_inv_iff G✝ : Type u_1 α : Type u_2 inst✝⁵ : Inv G✝ inst✝⁴ : MeasurableSpace G✝ inst✝³ : MeasurableInv G✝ m : MeasurableSpace α f✝ : α → G✝ μ : Measure α G : Type u_3 inst✝² : Group G inst✝¹ : MeasurableSpace G inst✝ : MeasurableInv G f : α → G h : Measurable fun x => (f x)⁻¹ ⊢ Measurable f simpa only [inv_inv] using h.inv ** Qed
aemeasurable_inv_iff G✝ : Type u_1 α : Type u_2 inst✝⁵ : Inv G✝ inst✝⁴ : MeasurableSpace G✝ inst✝³ : MeasurableInv G✝ m : MeasurableSpace α f✝ : α → G✝ μ : Measure α G : Type u_3 inst✝² : Group G inst✝¹ : MeasurableSpace G inst✝ : MeasurableInv G f : α → G h : AEMeasurable fun x => (f x)⁻¹ ⊢ AEMeasurable f simpa only [inv_inv] using h.inv ** Qed
measurable_inv_iff₀ G : Type u_1 α : Type u_2 inst✝⁵ : Inv G inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableInv G m : MeasurableSpace α f✝ : α → G μ : Measure α G₀ : Type u_3 inst✝² : GroupWithZero G₀ inst✝¹ : MeasurableSpace G₀ inst✝ : MeasurableInv G₀ f : α → G₀ h : Measurable fun x => (f x)⁻¹ ⊢ Measurable f simpa only [inv_inv] using h.inv ** Qed
aemeasurable_inv_iff₀ G : Type u_1 α : Type u_2 inst✝⁵ : Inv G inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableInv G m : MeasurableSpace α f✝ : α → G μ : Measure α G₀ : Type u_3 inst✝² : GroupWithZero G₀ inst✝¹ : MeasurableSpace G₀ inst✝ : MeasurableInv G₀ f : α → G₀ h : AEMeasurable fun x => (f x)⁻¹ ⊢ AEMeasurable f simpa only [inv_inv] using h.inv ** Qed
measurable_const_smul_iff M : Type u_1 β : Type u_2 α : Type u_3 inst✝⁹ : MeasurableSpace M inst✝⁸ : MeasurableSpace β inst✝⁷ : Monoid M inst✝⁶ : MulAction M β inst✝⁵ : MeasurableSMul M β inst✝⁴ : MeasurableSpace α f : α → β μ : Measure α G : Type u_4 inst✝³ : Group G inst✝² : MeasurableSpace G inst✝¹ : MulAction G β inst✝ : MeasurableSMul G β c : G h : Measurable fun x => c • f x ⊢ Measurable f simpa only [inv_smul_smul] using h.const_smul' c⁻¹ ** Qed
aemeasurable_const_smul_iff M : Type u_1 β : Type u_2 α : Type u_3 inst✝⁹ : MeasurableSpace M inst✝⁸ : MeasurableSpace β inst✝⁷ : Monoid M inst✝⁶ : MulAction M β inst✝⁵ : MeasurableSMul M β inst✝⁴ : MeasurableSpace α f : α → β μ : Measure α G : Type u_4 inst✝³ : Group G inst✝² : MeasurableSpace G inst✝¹ : MulAction G β inst✝ : MeasurableSMul G β c : G h : AEMeasurable fun x => c • f x ⊢ AEMeasurable f simpa only [inv_smul_smul] using h.const_smul' c⁻¹ ** Qed