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isPiSystem_image_Iio ** α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α s : Set α ⊢ IsPiSystem (Iio '' s) ** rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ - ** case intro.intro.intro.intro α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α s : Set α a : α ha : a ∈ s b : α hb : b ∈ s ⊢ Iio a ∩ Iio b ∈ Iio '' s ** exact ⟨a ⊓ b, inf_ind a b ha hb, Iio_inter_Iio.symm⟩ ** Qed
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isPiSystem_image_Iic ** α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α s : Set α ⊢ IsPiSystem (Iic '' s) ** rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ - ** case intro.intro.intro.intro α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α s : Set α a : α ha : a ∈ s b : α hb : b ∈ s ⊢ Iic a ∩ Iic b ∈ Iic '' s ** exact ⟨a ⊓ b, inf_ind a b ha hb, Iic_inter_Iic.symm⟩ ** Qed
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isPiSystem_Ixx_mem ** α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α Ixx : α → α → Set α p : α → α → Prop Hne : ∀ {a b : α}, Set.Nonempty (Ixx a b) → p a b Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂) s t : Set α ⊢ IsPiSystem {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ p l u ∧ Ixx l u = S} ** rintro _ ⟨l₁, hls₁, u₁, hut₁, _, rfl⟩ _ ⟨l₂, hls₂, u₂, hut₂, _, rfl⟩ ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α Ixx : α → α → Set α p : α → α → Prop Hne : ∀ {a b : α}, Set.Nonempty (Ixx a b) → p a b Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂) s t : Set α l₁ : α hls₁ : l₁ ∈ s u₁ : α hut₁ : u₁ ∈ t left✝¹ : p l₁ u₁ l₂ : α hls₂ : l₂ ∈ s u₂ : α hut₂ : u₂ ∈ t left✝ : p l₂ u₂ ⊢ Set.Nonempty (Ixx l₁ u₁ ∩ Ixx l₂ u₂) → Ixx l₁ u₁ ∩ Ixx l₂ u₂ ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ p l u ∧ Ixx l u = S} ** simp only [Hi] ** case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α Ixx : α → α → Set α p : α → α → Prop Hne : ∀ {a b : α}, Set.Nonempty (Ixx a b) → p a b Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂) s t : Set α l₁ : α hls₁ : l₁ ∈ s u₁ : α hut₁ : u₁ ∈ t left✝¹ : p l₁ u₁ l₂ : α hls₂ : l₂ ∈ s u₂ : α hut₂ : u₂ ∈ t left✝ : p l₂ u₂ ⊢ Set.Nonempty (Ixx (max l₁ l₂) (min u₁ u₂)) → Ixx (max l₁ l₂) (min u₁ u₂) ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ p l u ∧ Ixx l u = S} ** exact fun H => ⟨l₁ ⊔ l₂, sup_ind l₁ l₂ hls₁ hls₂, u₁ ⊓ u₂, inf_ind u₁ u₂ hut₁ hut₂, Hne H, rfl⟩ ** Qed
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isPiSystem_Ixx ** α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α Ixx : α → α → Set α p : α → α → Prop Hne : ∀ {a b : α}, Set.Nonempty (Ixx a b) → p a b Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂) f : ι → α g : ι' → α ⊢ IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S} ** simpa only [exists_range_iff] using isPiSystem_Ixx_mem (@Hne) (@Hi) (range f) (range g) ** Qed
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isPiSystem_Icc_mem ** α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α s t : Set α ⊢ ∀ {a₁ b₁ a₂ b₂ : α}, Icc a₁ b₁ ∩ Icc a₂ b₂ = Icc (max a₁ a₂) (min b₁ b₂) ** exact Icc_inter_Icc ** Qed
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isPiSystem_Icc ** α : Type u_1 ι : Sort u_2 ι' : Sort u_3 inst✝ : LinearOrder α f : ι → α g : ι' → α ⊢ ∀ {a₁ b₁ a₂ b₂ : α}, Icc a₁ b₁ ∩ Icc a₂ b₂ = Icc (max a₁ a₂) (min b₁ b₂) ** exact Icc_inter_Icc ** Qed
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generatePiSystem_subset_self ** α : Type u_1 S : Set (Set α) h_S : IsPiSystem S x : Set α h : x ∈ generatePiSystem S ⊢ x ∈ S ** induction' h with _ h_s s u _ _ h_nonempty h_s h_u ** case base α : Type u_1 S : Set (Set α) h_S : IsPiSystem S x s✝ : Set α h_s : s✝ ∈ S ⊢ s✝ ∈ S ** exact h_s ** case inter α : Type u_1 S : Set (Set α) h_S : IsPiSystem S x s u : Set α h_s✝ : generatePiSystem S s h_t✝ : generatePiSystem S u h_nonempty : Set.Nonempty (s ∩ u) h_s : s ∈ S h_u : u ∈ S ⊢ s ∩ u ∈ S ** exact h_S _ h_s _ h_u h_nonempty ** Qed
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generatePiSystem_measurableSet ** α : Type u_1 M : MeasurableSpace α S : Set (Set α) h_meas_S : ∀ (s : Set α), s ∈ S → MeasurableSet s t : Set α h_in_pi : t ∈ generatePiSystem S ⊢ MeasurableSet t ** induction' h_in_pi with s h_s s u _ _ _ h_s h_u ** case base α : Type u_1 M : MeasurableSpace α S : Set (Set α) h_meas_S : ∀ (s : Set α), s ∈ S → MeasurableSet s t s : Set α h_s : s ∈ S ⊢ MeasurableSet s ** apply h_meas_S _ h_s ** case inter α : Type u_1 M : MeasurableSpace α S : Set (Set α) h_meas_S : ∀ (s : Set α), s ∈ S → MeasurableSet s t s u : Set α h_s✝ : generatePiSystem S s h_t✝ : generatePiSystem S u h_nonempty✝ : Set.Nonempty (s ∩ u) h_s : MeasurableSet s h_u : MeasurableSet u ⊢ MeasurableSet (s ∩ u) ** apply MeasurableSet.inter h_s h_u ** Qed
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mem_generatePiSystem_iUnion_elim ** α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α h_t : t ∈ generatePiSystem (⋃ b, g b) ⊢ ∃ T f, t = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b ** induction' h_t with s h_s s t' h_gen_s h_gen_t' h_nonempty h_s h_t' ** case base α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t s : Set α h_s : s ∈ ⋃ b, g b ⊢ ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b ** rcases h_s with ⟨t', ⟨⟨b, rfl⟩, h_s_in_t'⟩⟩ ** case base.intro.intro.intro α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t s : Set α b : β h_s_in_t' : s ∈ (fun b => g b) b ⊢ ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b ** refine' ⟨{b}, fun _ => s, _⟩ ** case base.intro.intro.intro α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t s : Set α b : β h_s_in_t' : s ∈ (fun b => g b) b ⊢ s = ⋂ b_1 ∈ {b}, (fun x => s) b_1 ∧ ∀ (b_1 : β), b_1 ∈ {b} → (fun x => s) b_1 ∈ g b_1 ** simpa using h_s_in_t' ** case inter α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t s t' : Set α h_gen_s : generatePiSystem (⋃ b, g b) s h_gen_t' : generatePiSystem (⋃ b, g b) t' h_nonempty : Set.Nonempty (s ∩ t') h_s : ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b h_t' : ∃ T f, t' = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b ⊢ ∃ T f, s ∩ t' = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b ** rcases h_t' with ⟨T_t', ⟨f_t', ⟨rfl, h_t'⟩⟩⟩ ** case inter.intro.intro.intro α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t s : Set α h_gen_s : generatePiSystem (⋃ b, g b) s h_s : ∃ T f, s = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) h_nonempty : Set.Nonempty (s ∩ ⋂ b ∈ T_t', f_t' b) ⊢ ∃ T f, s ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b ** rcases h_s with ⟨T_s, ⟨f_s, ⟨rfl, h_s⟩⟩⟩ ** case inter.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) ⊢ ∃ T f, (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T, f b ∧ ∀ (b : β), b ∈ T → f b ∈ g b ** use T_s ∪ T_t', fun b : β =>
if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b
else if b ∈ T_t' then f_t' b else (∅ : Set α) ** case h α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) ⊢ ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∧ ∀ (b : β), b ∈ T_s ∪ T_t' → (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b ** constructor ** case h.right α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) ⊢ ∀ (b : β), b ∈ T_s ∪ T_t' → (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b ** intro b h_b ** case h.right α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) b : β h_b : b ∈ T_s ∪ T_t' ⊢ (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b ** split_ifs with hbs hbt hbt ** case h.left α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) ⊢ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b = ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅ ** ext a ** case h.left.h α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α ⊢ a ∈ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ↔ a ∈ ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅ ** simp_rw [Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union, or_imp] ** case h.left.h α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α ⊢ ((∀ (i : β), i ∈ T_s → a ∈ f_s i) ∧ ∀ (i : β), i ∈ T_t' → a ∈ f_t' i) ↔ ∀ (i : β), (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧ (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ** rw [← forall_and] ** case h.left.h α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α ⊢ (∀ (x : β), (x ∈ T_s → a ∈ f_s x) ∧ (x ∈ T_t' → a ∈ f_t' x)) ↔ ∀ (i : β), (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧ (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ** constructor <;> intro h1 b <;> by_cases hbs : b ∈ T_s <;> by_cases hbt : b ∈ T_t' <;>
specialize h1 b <;>
simp only [hbs, hbt, if_true, if_false, true_imp_iff, and_self_iff, false_imp_iff,
and_true_iff, true_and_iff] at h1 ⊢ ** case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α b : β hbs : b ∈ T_s hbt : b ∈ T_t' h1 : a ∈ f_s b ∧ a ∈ f_t' b ⊢ a ∈ f_s b ∩ f_t' b case neg α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α b : β hbs : b ∈ T_s hbt : ¬b ∈ T_t' h1 : a ∈ f_s b ⊢ a ∈ f_s b case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α b : β hbs : ¬b ∈ T_s hbt : b ∈ T_t' h1 : a ∈ f_t' b ⊢ a ∈ f_t' b case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α b : β hbs : b ∈ T_s hbt : b ∈ T_t' h1 : a ∈ f_s b ∩ f_t' b ⊢ a ∈ f_s b ∧ a ∈ f_t' b case neg α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α b : β hbs : b ∈ T_s hbt : ¬b ∈ T_t' h1 : a ∈ f_s b ⊢ a ∈ f_s b case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α b : β hbs : ¬b ∈ T_s hbt : b ∈ T_t' h1 : a ∈ f_t' b ⊢ a ∈ f_t' b ** all_goals exact h1 ** case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) a : α b : β hbs : ¬b ∈ T_s hbt : b ∈ T_t' h1 : a ∈ f_t' b ⊢ a ∈ f_t' b ** exact h1 ** case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) b : β h_b : b ∈ T_s ∪ T_t' hbs : b ∈ T_s hbt : b ∈ T_t' ⊢ f_s b ∩ f_t' b ∈ g b ** refine' h_pi b (f_s b) (h_s b hbs) (f_t' b) (h_t' b hbt) (Set.Nonempty.mono _ h_nonempty) ** case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) b : β h_b : b ∈ T_s ∪ T_t' hbs : b ∈ T_s hbt : b ∈ T_t' ⊢ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ⊆ f_s b ∩ f_t' b ** exact Set.inter_subset_inter (Set.biInter_subset_of_mem hbs) (Set.biInter_subset_of_mem hbt) ** case neg α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) b : β h_b : b ∈ T_s ∪ T_t' hbs : b ∈ T_s hbt : ¬b ∈ T_t' ⊢ f_s b ∈ g b ** exact h_s b hbs ** case pos α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) b : β h_b : b ∈ T_s ∪ T_t' hbs : ¬b ∈ T_s hbt : b ∈ T_t' ⊢ f_t' b ∈ g b ** exact h_t' b hbt ** case neg α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) b : β h_b : b ∈ T_s ∪ T_t' hbs : ¬b ∈ T_s hbt : ¬b ∈ T_t' ⊢ ∅ ∈ g b ** rw [Finset.mem_union] at h_b ** case neg α : Type u_1 β : Type u_2 g : β → Set (Set α) h_pi : ∀ (b : β), IsPiSystem (g b) t : Set α T_t' : Finset β f_t' : β → Set α h_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b) T_s : Finset β f_s : β → Set α h_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b) h_nonempty : Set.Nonempty ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b) b : β h_b : b ∈ T_s ∨ b ∈ T_t' hbs : ¬b ∈ T_s hbt : ¬b ∈ T_t' ⊢ ∅ ∈ g b ** apply False.elim (h_b.elim hbs hbt) ** Qed
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piiUnionInter_singleton ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι ⊢ piiUnionInter π {i} = π i ∪ {univ} ** ext1 s ** case h α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α ⊢ s ∈ piiUnionInter π {i} ↔ s ∈ π i ∪ {univ} ** simp only [piiUnionInter, exists_prop, mem_union] ** case h α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α ⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ x ∈ t, f x} ↔ s ∈ π i ∨ s ∈ {univ} ** refine' ⟨_, fun h => _⟩ ** case h.refine'_1 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α ⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ x ∈ t, f x} → s ∈ π i ∨ s ∈ {univ} ** rintro ⟨t, hti, f, hfπ, rfl⟩ ** case h.refine'_1.intro.intro.intro.intro α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι hti : ↑t ⊆ {i} f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x ⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ} ** simp only [subset_singleton_iff, Finset.mem_coe] at hti ** case h.refine'_1.intro.intro.intro.intro α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i ⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ} ** by_cases hi : i ∈ t ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : i ∈ t ⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ} ** have ht_eq_i : t = {i} := by
ext1 x
rw [Finset.mem_singleton]
exact ⟨fun h => hti x h, fun h => h.symm ▸ hi⟩ ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : i ∈ t ht_eq_i : t = {i} ⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ} ** simp only [ht_eq_i, Finset.mem_singleton, iInter_iInter_eq_left] ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : i ∈ t ht_eq_i : t = {i} ⊢ f i ∈ π i ∨ f i ∈ {univ} ** exact Or.inl (hfπ i hi) ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : i ∈ t ⊢ t = {i} ** ext1 x ** case a α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : i ∈ t x : ι ⊢ x ∈ t ↔ x ∈ {i} ** rw [Finset.mem_singleton] ** case a α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : i ∈ t x : ι ⊢ x ∈ t ↔ x = i ** exact ⟨fun h => hti x h, fun h => h.symm ▸ hi⟩ ** case neg α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : ¬i ∈ t ⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ} ** have ht_empty : t = ∅ := by
ext1 x
simp only [Finset.not_mem_empty, iff_false_iff]
exact fun hx => hi (hti x hx ▸ hx) ** case neg α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : ¬i ∈ t ht_empty : t = ∅ ⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ} ** simp [ht_empty, Finset.not_mem_empty, iInter_false, iInter_univ, Set.mem_singleton univ,
or_true_iff] ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : ¬i ∈ t ⊢ t = ∅ ** ext1 x ** case a α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : ¬i ∈ t x : ι ⊢ x ∈ t ↔ x ∈ ∅ ** simp only [Finset.not_mem_empty, iff_false_iff] ** case a α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι t : Finset ι f : ι → Set α hfπ : ∀ (x : ι), x ∈ t → f x ∈ π x hti : ∀ (y : ι), y ∈ t → y = i hi : ¬i ∈ t x : ι ⊢ ¬x ∈ t ** exact fun hx => hi (hti x hx ▸ hx) ** case h.refine'_2 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α h : s ∈ π i ∨ s ∈ {univ} ⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ x ∈ t, f x} ** cases' h with hs hs ** case h.refine'_2.inl α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α hs : s ∈ π i ⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ x ∈ t, f x} ** refine' ⟨{i}, _, fun _ => s, ⟨fun x hx => _, _⟩⟩ ** case h.refine'_2.inl.refine'_1 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α hs : s ∈ π i ⊢ ↑{i} ⊆ {i} ** rw [Finset.coe_singleton] ** case h.refine'_2.inl.refine'_2 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α hs : s ∈ π i x : ι hx : x ∈ {i} ⊢ (fun x => s) x ∈ π x ** rw [Finset.mem_singleton] at hx ** case h.refine'_2.inl.refine'_2 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α hs : s ∈ π i x : ι hx : x = i ⊢ (fun x => s) x ∈ π x ** rwa [hx] ** case h.refine'_2.inl.refine'_3 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α hs : s ∈ π i ⊢ s = ⋂ x ∈ {i}, (fun x => s) x ** simp only [Finset.mem_singleton, iInter_iInter_eq_left] ** case h.refine'_2.inr α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α hs : s ∈ {univ} ⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ x ∈ t, f x} ** refine' ⟨∅, _⟩ ** case h.refine'_2.inr α : Type u_1 ι : Type u_2 π : ι → Set (Set α) i : ι s : Set α hs : s ∈ {univ} ⊢ ↑∅ ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ ∅ → f x ∈ π x) ∧ s = ⋂ x ∈ ∅, f x ** simpa only [Finset.coe_empty, subset_singleton_iff, mem_empty_iff_false, IsEmpty.forall_iff,
imp_true_iff, Finset.not_mem_empty, iInter_false, iInter_univ, true_and_iff,
exists_const] using hs ** Qed
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piiUnionInter_singleton_left ** α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι ⊢ piiUnionInter (fun i => {s i}) S = {s' | ∃ t x, s' = ⋂ i ∈ t, s i} ** ext1 s' ** case h α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι s' : Set α ⊢ s' ∈ piiUnionInter (fun i => {s i}) S ↔ s' ∈ {s' | ∃ t x, s' = ⋂ i ∈ t, s i} ** simp_rw [piiUnionInter, Set.mem_singleton_iff, exists_prop, Set.mem_setOf_eq] ** case h α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι s' : Set α ⊢ (∃ t, ↑t ⊆ S ∧ ∃ f, (∀ (x : ι), x ∈ t → f x = s x) ∧ s' = ⋂ x ∈ t, f x) ↔ ∃ t, ↑t ⊆ S ∧ s' = ⋂ i ∈ t, s i ** refine' ⟨fun h => _, fun ⟨t, htS, h_eq⟩ => ⟨t, htS, s, fun _ _ => rfl, h_eq⟩⟩ ** case h α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι s' : Set α h : ∃ t, ↑t ⊆ S ∧ ∃ f, (∀ (x : ι), x ∈ t → f x = s x) ∧ s' = ⋂ x ∈ t, f x ⊢ ∃ t, ↑t ⊆ S ∧ s' = ⋂ i ∈ t, s i ** obtain ⟨t, htS, f, hft_eq, rfl⟩ := h ** case h.intro.intro.intro.intro α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι t : Finset ι htS : ↑t ⊆ S f : ι → Set α hft_eq : ∀ (x : ι), x ∈ t → f x = s x ⊢ ∃ t_1, ↑t_1 ⊆ S ∧ ⋂ x ∈ t, f x = ⋂ i ∈ t_1, s i ** refine' ⟨t, htS, _⟩ ** case h.intro.intro.intro.intro α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι t : Finset ι htS : ↑t ⊆ S f : ι → Set α hft_eq : ∀ (x : ι), x ∈ t → f x = s x ⊢ ⋂ x ∈ t, f x = ⋂ i ∈ t, s i ** congr! 3 ** case h.intro.intro.intro.intro.h.e'_3.h.f α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι t : Finset ι htS : ↑t ⊆ S f : ι → Set α hft_eq : ∀ (x : ι), x ∈ t → f x = s x x✝¹ : ι x✝ : x✝¹ ∈ t ⊢ f x✝¹ = s x✝¹ ** apply hft_eq ** case h.intro.intro.intro.intro.h.e'_3.h.f.a α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι t : Finset ι htS : ↑t ⊆ S f : ι → Set α hft_eq : ∀ (x : ι), x ∈ t → f x = s x x✝¹ : ι x✝ : x✝¹ ∈ t ⊢ x✝¹ ∈ t ** assumption ** Qed
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generateFrom_piiUnionInter_singleton_left ** α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι ⊢ generateFrom (piiUnionInter (fun k => {s k}) S) = generateFrom {t | ∃ k, k ∈ S ∧ s k = t} ** refine' le_antisymm (generateFrom_le _) (generateFrom_mono _) ** case refine'_1 α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι ⊢ ∀ (t : Set α), t ∈ piiUnionInter (fun k => {s k}) S → MeasurableSet t ** rintro _ ⟨I, hI, f, hf, rfl⟩ ** case refine'_1.intro.intro.intro.intro α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι I : Finset ι hI : ↑I ⊆ S f : ι → Set α hf : ∀ (x : ι), x ∈ I → f x ∈ (fun k => {s k}) x ⊢ MeasurableSet (⋂ x ∈ I, f x) ** refine' Finset.measurableSet_biInter _ fun m hm => measurableSet_generateFrom _ ** case refine'_1.intro.intro.intro.intro α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι I : Finset ι hI : ↑I ⊆ S f : ι → Set α hf : ∀ (x : ι), x ∈ I → f x ∈ (fun k => {s k}) x m : ι hm : m ∈ I ⊢ f m ∈ {t | ∃ k, k ∈ S ∧ s k = t} ** exact ⟨m, hI hm, (hf m hm).symm⟩ ** case refine'_2 α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι ⊢ {t | ∃ k, k ∈ S ∧ s k = t} ⊆ piiUnionInter (fun k => {s k}) S ** rintro _ ⟨k, hk, rfl⟩ ** case refine'_2.intro.intro α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι k : ι hk : k ∈ S ⊢ s k ∈ piiUnionInter (fun k => {s k}) S ** refine' ⟨{k}, fun m hm => _, s, fun i _ => _, _⟩ ** case refine'_2.intro.intro.refine'_1 α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι k : ι hk : k ∈ S m : ι hm : m ∈ ↑{k} ⊢ m ∈ S ** rw [Finset.mem_coe, Finset.mem_singleton] at hm ** case refine'_2.intro.intro.refine'_1 α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι k : ι hk : k ∈ S m : ι hm : m = k ⊢ m ∈ S ** rwa [hm] ** case refine'_2.intro.intro.refine'_2 α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι k : ι hk : k ∈ S i : ι x✝ : i ∈ {k} ⊢ s i ∈ (fun k => {s k}) i ** exact Set.mem_singleton _ ** case refine'_2.intro.intro.refine'_3 α : Type u_1 ι : Type u_2 s : ι → Set α S : Set ι k : ι hk : k ∈ S ⊢ s k = ⋂ x ∈ {k}, s x ** simp only [Finset.mem_singleton, Set.iInter_iInter_eq_left] ** Qed
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isPiSystem_piiUnionInter ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι ⊢ IsPiSystem (piiUnionInter π S) ** rintro t1 ⟨p1, hp1S, f1, hf1m, ht1_eq⟩ t2 ⟨p2, hp2S, f2, hf2m, ht2_eq⟩ h_nonempty ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) ⊢ t1 ∩ t2 ∈ piiUnionInter π S ** simp_rw [piiUnionInter, Set.mem_setOf_eq] ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) ⊢ ∃ t x f x, t1 ∩ t2 = ⋂ x ∈ t, f x ** let g n := ite (n ∈ p1) (f1 n) Set.univ ∩ ite (n ∈ p2) (f2 n) Set.univ ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ ⊢ ∃ t x f x, t1 ∩ t2 = ⋂ x ∈ t, f x ** have hp_union_ss : ↑(p1 ∪ p2) ⊆ S := by
simp only [hp1S, hp2S, Finset.coe_union, union_subset_iff, and_self_iff] ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S ⊢ ∃ t x f x, t1 ∩ t2 = ⋂ x ∈ t, f x ** use p1 ∪ p2, hp_union_ss, g ** case h α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i ⊢ ∃ x, t1 ∩ t2 = ⋂ x ∈ p1 ∪ p2, g x ** refine' ⟨fun n hn => _, h_inter_eq⟩ ** case h α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 ⊢ g n ∈ π n ** simp only [] ** case h α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 ⊢ ((if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ) ∈ π n ** split_ifs with hn1 hn2 h ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ ⊢ ↑(p1 ∪ p2) ⊆ S ** simp only [hp1S, hp2S, Finset.coe_union, union_subset_iff, and_self_iff] ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S ⊢ t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i ** rw [ht1_eq, ht2_eq] ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S ⊢ (⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x = ⋂ i ∈ p1 ∪ p2, g i ** simp_rw [← Set.inf_eq_inter] ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S ⊢ (⋂ x ∈ p1, f1 x) ⊓ ⋂ x ∈ p2, f2 x = ⋂ i ∈ p1 ∪ p2, (if i ∈ p1 then f1 i else univ) ⊓ if i ∈ p2 then f2 i else univ ** ext1 x ** case h α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α ⊢ x ∈ (⋂ x ∈ p1, f1 x) ⊓ ⋂ x ∈ p2, f2 x ↔ x ∈ ⋂ i ∈ p1 ∪ p2, (if i ∈ p1 then f1 i else univ) ⊓ if i ∈ p2 then f2 i else univ ** simp only [inf_eq_inter, mem_inter_iff, mem_iInter, Finset.mem_union] ** case h α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α ⊢ ((∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i) ↔ ∀ (i : ι), i ∈ p1 ∨ i ∈ p2 → (x ∈ if i ∈ p1 then f1 i else univ) ∧ x ∈ if i ∈ p2 then f2 i else univ ** refine' ⟨fun h i _ => _, fun h => ⟨fun i hi1 => _, fun i hi2 => _⟩⟩ ** case h.refine'_1 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α h : (∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i i : ι x✝ : i ∈ p1 ∨ i ∈ p2 ⊢ (x ∈ if i ∈ p1 then f1 i else univ) ∧ x ∈ if i ∈ p2 then f2 i else univ ** split_ifs with h_1 h_2 h_2 ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α h : (∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i i : ι x✝ : i ∈ p1 ∨ i ∈ p2 h_1 : i ∈ p1 h_2 : i ∈ p2 ⊢ x ∈ f1 i ∧ x ∈ f2 i case neg α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α h : (∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i i : ι x✝ : i ∈ p1 ∨ i ∈ p2 h_1 : i ∈ p1 h_2 : ¬i ∈ p2 ⊢ x ∈ f1 i ∧ x ∈ univ case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α h : (∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i i : ι x✝ : i ∈ p1 ∨ i ∈ p2 h_1 : ¬i ∈ p1 h_2 : i ∈ p2 ⊢ x ∈ univ ∧ x ∈ f2 i case neg α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α h : (∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i i : ι x✝ : i ∈ p1 ∨ i ∈ p2 h_1 : ¬i ∈ p1 h_2 : ¬i ∈ p2 ⊢ x ∈ univ ∧ x ∈ univ ** exacts [⟨h.1 i h_1, h.2 i h_2⟩, ⟨h.1 i h_1, Set.mem_univ _⟩, ⟨Set.mem_univ _, h.2 i h_2⟩,
⟨Set.mem_univ _, Set.mem_univ _⟩] ** case h.refine'_2 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α h : ∀ (i : ι), i ∈ p1 ∨ i ∈ p2 → (x ∈ if i ∈ p1 then f1 i else univ) ∧ x ∈ if i ∈ p2 then f2 i else univ i : ι hi1 : i ∈ p1 ⊢ x ∈ f1 i ** specialize h i (Or.inl hi1) ** case h.refine'_2 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α i : ι hi1 : i ∈ p1 h : (x ∈ if i ∈ p1 then f1 i else univ) ∧ x ∈ if i ∈ p2 then f2 i else univ ⊢ x ∈ f1 i ** rw [if_pos hi1] at h ** case h.refine'_2 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α i : ι hi1 : i ∈ p1 h : x ∈ f1 i ∧ x ∈ if i ∈ p2 then f2 i else univ ⊢ x ∈ f1 i ** exact h.1 ** case h.refine'_3 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α h : ∀ (i : ι), i ∈ p1 ∨ i ∈ p2 → (x ∈ if i ∈ p1 then f1 i else univ) ∧ x ∈ if i ∈ p2 then f2 i else univ i : ι hi2 : i ∈ p2 ⊢ x ∈ f2 i ** specialize h i (Or.inr hi2) ** case h.refine'_3 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α i : ι hi2 : i ∈ p2 h : (x ∈ if i ∈ p1 then f1 i else univ) ∧ x ∈ if i ∈ p2 then f2 i else univ ⊢ x ∈ f2 i ** rw [if_pos hi2] at h ** case h.refine'_3 α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S x : α i : ι hi2 : i ∈ p2 h : (x ∈ if i ∈ p1 then f1 i else univ) ∧ x ∈ f2 i ⊢ x ∈ f2 i ** exact h.2 ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 ⊢ f1 n ∩ f2 n ∈ π n ** refine' hpi n (f1 n) (hf1m n hn1) (f2 n) (hf2m n hn2) (Set.nonempty_iff_ne_empty.2 fun h => _) ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ ⊢ False ** rw [h_inter_eq] at h_nonempty ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ h_nonempty : Set.Nonempty (⋂ i ∈ p1 ∪ p2, g i) hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ ⊢ False ** suffices h_empty : ⋂ i ∈ p1 ∪ p2, g i = ∅ ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ h_nonempty : Set.Nonempty (⋂ i ∈ p1 ∪ p2, g i) hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ h_empty : ⋂ i ∈ p1 ∪ p2, g i = ∅ ⊢ False case h_empty α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ h_nonempty : Set.Nonempty (⋂ i ∈ p1 ∪ p2, g i) hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ ⊢ ⋂ i ∈ p1 ∪ p2, g i = ∅ ** exact (Set.not_nonempty_iff_eq_empty.mpr h_empty) h_nonempty ** case h_empty α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ h_nonempty : Set.Nonempty (⋂ i ∈ p1 ∪ p2, g i) hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ ⊢ ⋂ i ∈ p1 ∪ p2, g i = ∅ ** refine' le_antisymm (Set.iInter_subset_of_subset n _) (Set.empty_subset _) ** case h_empty α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ h_nonempty : Set.Nonempty (⋂ i ∈ p1 ∪ p2, g i) hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ ⊢ ⋂ (_ : n ∈ p1 ∪ p2), g n ⊆ ∅ ** refine' Set.iInter_subset_of_subset hn _ ** case h_empty α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ h_nonempty : Set.Nonempty (⋂ i ∈ p1 ∪ p2, g i) hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ ⊢ g n ⊆ ∅ ** simp_rw [if_pos hn1, if_pos hn2] ** case h_empty α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ h_nonempty : Set.Nonempty (⋂ i ∈ p1 ∪ p2, g i) hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : n ∈ p2 h : f1 n ∩ f2 n = ∅ ⊢ f1 n ∩ f2 n ⊆ ∅ ** exact h.subset ** case neg α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : n ∈ p1 hn2 : ¬n ∈ p2 ⊢ f1 n ∩ univ ∈ π n ** simp [hf1m n hn1] ** case pos α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : ¬n ∈ p1 h : n ∈ p2 ⊢ univ ∩ f2 n ∈ π n ** simp [hf2m n h] ** case neg α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : ¬n ∈ p1 h : ¬n ∈ p2 ⊢ univ ∩ univ ∈ π n ** exact absurd hn (by simp [hn1, h]) ** α : Type u_1 ι : Type u_2 π : ι → Set (Set α) hpi : ∀ (x : ι), IsPiSystem (π x) S : Set ι t1 : Set α p1 : Finset ι hp1S : ↑p1 ⊆ S f1 : ι → Set α hf1m : ∀ (x : ι), x ∈ p1 → f1 x ∈ π x ht1_eq : t1 = ⋂ x ∈ p1, f1 x t2 : Set α p2 : Finset ι hp2S : ↑p2 ⊆ S f2 : ι → Set α hf2m : ∀ (x : ι), x ∈ p2 → f2 x ∈ π x ht2_eq : t2 = ⋂ x ∈ p2, f2 x h_nonempty : Set.Nonempty (t1 ∩ t2) g : ι → Set α := fun n => (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ hp_union_ss : ↑(p1 ∪ p2) ⊆ S h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i n : ι hn : n ∈ p1 ∪ p2 hn1 : ¬n ∈ p1 h : ¬n ∈ p2 ⊢ ¬n ∈ p1 ∪ p2 ** simp [hn1, h] ** Qed
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generateFrom_piiUnionInter_le ** α : Type u_1 ι : Type u_2 m : MeasurableSpace α π : ι → Set (Set α) h : ∀ (n : ι), generateFrom (π n) ≤ m S : Set ι ⊢ generateFrom (piiUnionInter π S) ≤ m ** refine' generateFrom_le _ ** α : Type u_1 ι : Type u_2 m : MeasurableSpace α π : ι → Set (Set α) h : ∀ (n : ι), generateFrom (π n) ≤ m S : Set ι ⊢ ∀ (t : Set α), t ∈ piiUnionInter π S → MeasurableSet t ** rintro t ⟨ht_p, _, ft, hft_mem_pi, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 ι : Type u_2 m : MeasurableSpace α π : ι → Set (Set α) h : ∀ (n : ι), generateFrom (π n) ≤ m S : Set ι ht_p : Finset ι w✝ : ↑ht_p ⊆ S ft : ι → Set α hft_mem_pi : ∀ (x : ι), x ∈ ht_p → ft x ∈ π x ⊢ MeasurableSet (⋂ x ∈ ht_p, ft x) ** refine' Finset.measurableSet_biInter _ fun x hx_mem => (h x) _ _ ** case intro.intro.intro.intro α : Type u_1 ι : Type u_2 m : MeasurableSpace α π : ι → Set (Set α) h : ∀ (n : ι), generateFrom (π n) ≤ m S : Set ι ht_p : Finset ι w✝ : ↑ht_p ⊆ S ft : ι → Set α hft_mem_pi : ∀ (x : ι), x ∈ ht_p → ft x ∈ π x x : ι hx_mem : x ∈ ht_p ⊢ MeasurableSet (ft x) ** exact measurableSet_generateFrom (hft_mem_pi x hx_mem) ** Qed
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measurableSet_iSup_of_mem_piiUnionInter ** α : Type u_1 ι : Type u_2 m : ι → MeasurableSpace α S : Set ι t : Set α ht : t ∈ piiUnionInter (fun n => {s | MeasurableSet s}) S ⊢ MeasurableSet t ** rcases ht with ⟨pt, hpt, ft, ht_m, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 ι : Type u_2 m : ι → MeasurableSpace α S : Set ι pt : Finset ι hpt : ↑pt ⊆ S ft : ι → Set α ht_m : ∀ (x : ι), x ∈ pt → ft x ∈ (fun n => {s | MeasurableSet s}) x ⊢ MeasurableSet (⋂ x ∈ pt, ft x) ** refine' pt.measurableSet_biInter fun i hi => _ ** case intro.intro.intro.intro α : Type u_1 ι : Type u_2 m : ι → MeasurableSpace α S : Set ι pt : Finset ι hpt : ↑pt ⊆ S ft : ι → Set α ht_m : ∀ (x : ι), x ∈ pt → ft x ∈ (fun n => {s | MeasurableSet s}) x i : ι hi : i ∈ pt ⊢ MeasurableSet (ft i) ** suffices h_le : m i ≤ ⨆ i ∈ S, m i ** case intro.intro.intro.intro α : Type u_1 ι : Type u_2 m : ι → MeasurableSpace α S : Set ι pt : Finset ι hpt : ↑pt ⊆ S ft : ι → Set α ht_m : ∀ (x : ι), x ∈ pt → ft x ∈ (fun n => {s | MeasurableSet s}) x i : ι hi : i ∈ pt h_le : m i ≤ ⨆ i ∈ S, m i ⊢ MeasurableSet (ft i) case h_le α : Type u_1 ι : Type u_2 m : ι → MeasurableSpace α S : Set ι pt : Finset ι hpt : ↑pt ⊆ S ft : ι → Set α ht_m : ∀ (x : ι), x ∈ pt → ft x ∈ (fun n => {s | MeasurableSet s}) x i : ι hi : i ∈ pt ⊢ m i ≤ ⨆ i ∈ S, m i ** exact h_le (ft i) (ht_m i hi) ** case h_le α : Type u_1 ι : Type u_2 m : ι → MeasurableSpace α S : Set ι pt : Finset ι hpt : ↑pt ⊆ S ft : ι → Set α ht_m : ∀ (x : ι), x ∈ pt → ft x ∈ (fun n => {s | MeasurableSet s}) x i : ι hi : i ∈ pt ⊢ m i ≤ ⨆ i ∈ S, m i ** have hi' : i ∈ S := hpt hi ** case h_le α : Type u_1 ι : Type u_2 m : ι → MeasurableSpace α S : Set ι pt : Finset ι hpt : ↑pt ⊆ S ft : ι → Set α ht_m : ∀ (x : ι), x ∈ pt → ft x ∈ (fun n => {s | MeasurableSet s}) x i : ι hi : i ∈ pt hi' : i ∈ S ⊢ m i ≤ ⨆ i ∈ S, m i ** exact le_iSup₂ (f := fun i (_ : i ∈ S) => m i) i hi' ** Qed
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MeasurableSpace.DynkinSystem.ext ** α : Type u_1 s₁ : Set α → Prop has_empty✝¹ : s₁ ∅ has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i) s₂ : Set α → Prop has_empty✝ : s₂ ∅ has_compl✝ : ∀ {a : Set α}, s₂ a → s₂ aᶜ has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₂ (f i)) → s₂ (⋃ i, f i) h : ∀ (s : Set α), Has { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } s ↔ Has { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ } s ⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } = { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ } ** have : s₁ = s₂ := funext fun x => propext <| h x ** α : Type u_1 s₁ : Set α → Prop has_empty✝¹ : s₁ ∅ has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i) s₂ : Set α → Prop has_empty✝ : s₂ ∅ has_compl✝ : ∀ {a : Set α}, s₂ a → s₂ aᶜ has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₂ (f i)) → s₂ (⋃ i, f i) h : ∀ (s : Set α), Has { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } s ↔ Has { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ } s this : s₁ = s₂ ⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } = { Has := s₂, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ } ** subst this ** α : Type u_1 s₁ : Set α → Prop has_empty✝¹ : s₁ ∅ has_compl✝¹ : ∀ {a : Set α}, s₁ a → s₁ aᶜ has_iUnion_nat✝¹ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i) has_empty✝ : s₁ ∅ has_compl✝ : ∀ {a : Set α}, s₁ a → s₁ aᶜ has_iUnion_nat✝ : ∀ {f : ℕ → Set α}, Pairwise (Disjoint on f) → (∀ (i : ℕ), s₁ (f i)) → s₁ (⋃ i, f i) h : ∀ (s : Set α), Has { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } s ↔ Has { Has := s₁, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ } s ⊢ { Has := s₁, has_empty := has_empty✝¹, has_compl := has_compl✝¹, has_iUnion_nat := has_iUnion_nat✝¹ } = { Has := s₁, has_empty := has_empty✝, has_compl := has_compl✝, has_iUnion_nat := has_iUnion_nat✝ } ** rfl ** Qed
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MeasurableSpace.DynkinSystem.has_compl_iff ** α : Type u_1 d : DynkinSystem α a : Set α h : Has d aᶜ ⊢ Has d a ** simpa using d.has_compl h ** Qed
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MeasurableSpace.DynkinSystem.has_univ ** α : Type u_1 d : DynkinSystem α ⊢ Has d univ ** simpa using d.has_compl d.has_empty ** Qed
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MeasurableSpace.DynkinSystem.has_iUnion ** α : Type u_1 d : DynkinSystem α β : Type u_2 inst✝ : Countable β f : β → Set α hd : Pairwise (Disjoint on f) h : ∀ (i : β), Has d (f i) ⊢ Has d (⋃ i, f i) ** cases nonempty_encodable β ** case intro α : Type u_1 d : DynkinSystem α β : Type u_2 inst✝ : Countable β f : β → Set α hd : Pairwise (Disjoint on f) h : ∀ (i : β), Has d (f i) val✝ : Encodable β ⊢ Has d (⋃ i, f i) ** rw [← Encodable.iUnion_decode₂] ** case intro α : Type u_1 d : DynkinSystem α β : Type u_2 inst✝ : Countable β f : β → Set α hd : Pairwise (Disjoint on f) h : ∀ (i : β), Has d (f i) val✝ : Encodable β ⊢ Has d (⋃ i, ⋃ b ∈ Encodable.decode₂ β i, f b) ** exact
d.has_iUnion_nat (Encodable.iUnion_decode₂_disjoint_on hd) fun n =>
Encodable.iUnion_decode₂_cases d.has_empty h ** Qed
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MeasurableSpace.DynkinSystem.has_union ** α : Type u_1 d : DynkinSystem α s₁ s₂ : Set α h₁ : Has d s₁ h₂ : Has d s₂ h : Disjoint s₁ s₂ ⊢ Has d (s₁ ∪ s₂) ** rw [union_eq_iUnion] ** α : Type u_1 d : DynkinSystem α s₁ s₂ : Set α h₁ : Has d s₁ h₂ : Has d s₂ h : Disjoint s₁ s₂ ⊢ Has d (⋃ b, bif b then s₁ else s₂) ** exact d.has_iUnion (pairwise_disjoint_on_bool.2 h) (Bool.forall_bool.2 ⟨h₂, h₁⟩) ** Qed
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MeasurableSpace.DynkinSystem.has_diff ** α : Type u_1 d : DynkinSystem α s₁ s₂ : Set α h₁ : Has d s₁ h₂ : Has d s₂ h : s₂ ⊆ s₁ ⊢ Has d (s₁ \ s₂) ** apply d.has_compl_iff.1 ** α : Type u_1 d : DynkinSystem α s₁ s₂ : Set α h₁ : Has d s₁ h₂ : Has d s₂ h : s₂ ⊆ s₁ ⊢ Has d (s₁ \ s₂)ᶜ ** simp only [diff_eq, compl_inter, compl_compl] ** α : Type u_1 d : DynkinSystem α s₁ s₂ : Set α h₁ : Has d s₁ h₂ : Has d s₂ h : s₂ ⊆ s₁ ⊢ Has d (s₁ᶜ ∪ s₂) ** exact d.has_union (d.has_compl h₁) h₂ (disjoint_compl_left.mono_right h) ** Qed
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MeasurableSpace.DynkinSystem.generateHas_compl ** α : Type u_1 d : DynkinSystem α C : Set (Set α) s : Set α ⊢ GenerateHas C sᶜ ↔ GenerateHas C s ** refine' ⟨_, GenerateHas.compl⟩ ** α : Type u_1 d : DynkinSystem α C : Set (Set α) s : Set α ⊢ GenerateHas C sᶜ → GenerateHas C s ** intro h ** α : Type u_1 d : DynkinSystem α C : Set (Set α) s : Set α h : GenerateHas C sᶜ ⊢ GenerateHas C s ** convert GenerateHas.compl h ** case h.e'_3 α : Type u_1 d : DynkinSystem α C : Set (Set α) s : Set α h : GenerateHas C sᶜ ⊢ s = sᶜᶜ ** simp ** Qed
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blimsup_thickening_mul_ae_eq_aux ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ hr : Tendsto r atTop (𝓝 0) hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i ⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p ** have h₁ := blimsup_cthickening_ae_eq_blimsup_thickening (s := s) μ hr hr' ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ hr : Tendsto r atTop (𝓝 0) hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i h₁ : (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i ⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p ** have h₂ := blimsup_cthickening_mul_ae_eq μ p s hM r hr ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ hr : Tendsto r atTop (𝓝 0) hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i h₁ : (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p ⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p ** replace hr : Tendsto (fun i => M * r i) atTop (𝓝 0) ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i h₁ : (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p hr : Tendsto (fun i => M * r i) atTop (𝓝 0) ⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p ** replace hr' : ∀ᶠ i in atTop, p i → 0 < M * r i := hr'.mono fun i hi hip => mul_pos hM (hi hip) ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ h₁ : (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p hr : Tendsto (fun i => M * r i) atTop (𝓝 0) hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i ⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p ** have h₃ := blimsup_cthickening_ae_eq_blimsup_thickening (s := s) μ hr hr' ** α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ h₁ : (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p hr : Tendsto (fun i => M * r i) atTop (𝓝 0) hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i h₃ : (blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (M * r i) (s i)) atTop fun i => p i ⊢ blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop p ** exact h₃.symm.trans (h₂.trans h₁) ** case hr α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ hr : Tendsto r atTop (𝓝 0) hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i h₁ : (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p ⊢ Tendsto (fun i => M * r i) atTop (𝓝 0) ** convert hr.const_mul M ** case h.e'_5.h.e'_3 α : Type u_1 inst✝⁵ : MetricSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : MeasurableSpace α inst✝² : BorelSpace α μ : Measure α inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsUnifLocDoublingMeasure μ p : ℕ → Prop s : ℕ → Set α M : ℝ hM : 0 < M r : ℕ → ℝ hr : Tendsto r atTop (𝓝 0) hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i h₁ : (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i) =ᵐ[μ] blimsup (fun i => thickening (r i) (s i)) atTop fun i => p i h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p ⊢ 0 = M * 0 ** simp ** Qed
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Real.Icc_mem_vitaliFamily_at_right ** x y : ℝ hxy : x < y ⊢ Icc x y ∈ VitaliFamily.setsAt (vitaliFamily volume 1) x ** rw [Icc_eq_closedBall] ** x y : ℝ hxy : x < y ⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (vitaliFamily volume 1) x ** refine' closedBall_mem_vitaliFamily_of_dist_le_mul _ _ (by linarith) ** x y : ℝ hxy : x < y ⊢ dist x ((x + y) / 2) ≤ 1 * ((y - x) / 2) ** rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith ** x y : ℝ hxy : x < y ⊢ 0 < (y - x) / 2 ** linarith ** Qed
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Real.Icc_mem_vitaliFamily_at_left ** x y : ℝ hxy : x < y ⊢ Icc x y ∈ VitaliFamily.setsAt (vitaliFamily volume 1) y ** rw [Icc_eq_closedBall] ** x y : ℝ hxy : x < y ⊢ Metric.closedBall ((x + y) / 2) ((y - x) / 2) ∈ VitaliFamily.setsAt (vitaliFamily volume 1) y ** refine' closedBall_mem_vitaliFamily_of_dist_le_mul _ _ (by linarith) ** x y : ℝ hxy : x < y ⊢ dist y ((x + y) / 2) ≤ 1 * ((y - x) / 2) ** rw [Real.dist_eq, abs_of_nonneg] <;> linarith ** x y : ℝ hxy : x < y ⊢ 0 < (y - x) / 2 ** linarith ** Qed
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MeasureTheory.q ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : G ⊢ Memℒp (↑↑(condexpIndSMul hm hs hμs x)) 1 ** rw [memℒp_one_iff_integrable] ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : G ⊢ Integrable ↑↑(condexpIndSMul hm hs hμs x) ** apply integrable_condexpIndSMul ** Qed
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MeasureTheory.condexpIndL1_of_measurableSet_of_measure_ne_top ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : G ⊢ condexpIndL1 hm μ s x = condexpIndL1Fin hm hs hμs x ** simp only [condexpIndL1, And.intro hs hμs, dif_pos, Ne.def, not_false_iff, and_self_iff] ** Qed
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MeasureTheory.condexpIndL1_of_measure_eq_top ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hμs : ↑↑μ s = ⊤ x : G ⊢ condexpIndL1 hm μ s x = 0 ** simp only [condexpIndL1, hμs, eq_self_iff_true, not_true, Ne.def, dif_neg, not_false_iff,
and_false_iff] ** Qed
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MeasureTheory.condexpIndL1_of_not_measurableSet ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : ¬MeasurableSet s x : G ⊢ condexpIndL1 hm μ s x = 0 ** simp only [condexpIndL1, hs, dif_neg, not_false_iff, false_and_iff] ** Qed
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MeasureTheory.condexpInd_ae_eq_condexpIndSMul ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : G ⊢ ↑↑(↑(condexpInd G hm μ s) x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x) ** refine' EventuallyEq.trans _ (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x) ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : G ⊢ ↑↑(↑(condexpInd G hm μ s) x) =ᵐ[μ] ↑↑(condexpIndL1Fin hm hs hμs x) ** simp [condexpInd, condexpIndL1, hs, hμs] ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : G ⊢ ↑↑(condexpIndL1Fin hm (_ : MeasurableSet s) (_ : ↑↑μ s ≠ ⊤) x) =ᵐ[μ] ↑↑(condexpIndL1Fin hm hs hμs x) ** rfl ** Qed
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MeasureTheory.condexpInd_of_measurable ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G ⊢ ↑(condexpInd G hm μ s) c = indicatorConstLp 1 (_ : MeasurableSet s) hμs c ** ext1 ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G ⊢ ↑↑(↑(condexpInd G hm μ s) c) =ᵐ[μ] ↑↑(indicatorConstLp 1 (_ : MeasurableSet s) hμs c) ** refine' EventuallyEq.trans _ indicatorConstLp_coeFn.symm ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G ⊢ ↑↑(↑(condexpInd G hm μ s) c) =ᵐ[μ] Set.indicator s fun x => c ** refine' (condexpInd_ae_eq_condexpIndSMul hm (hm s hs) hμs c).trans _ ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G ⊢ ↑↑(condexpIndSMul hm (_ : MeasurableSet s) hμs c) =ᵐ[μ] Set.indicator s fun x => c ** refine' (condexpIndSMul_ae_eq_smul hm (hm s hs) hμs c).trans _ ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G ⊢ (fun a => ↑↑↑(↑(condexpL2 ℝ ℝ hm) (indicatorConstLp 2 (_ : MeasurableSet s) hμs 1)) a • c) =ᵐ[μ] Set.indicator s fun x => c ** rw [lpMeas_coe, condexpL2_indicator_of_measurable hm hs hμs (1 : ℝ)] ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G ⊢ (fun a => ↑↑(indicatorConstLp 2 (_ : MeasurableSet s) hμs 1) a • c) =ᵐ[μ] Set.indicator s fun x => c ** refine' (@indicatorConstLp_coeFn α _ _ 2 μ _ s (hm s hs) hμs (1 : ℝ)).mono fun x hx => _ ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G x : α hx : ↑↑(indicatorConstLp 2 (_ : MeasurableSet s) hμs 1) x = Set.indicator s (fun x => 1) x ⊢ (fun a => ↑↑(indicatorConstLp 2 (_ : MeasurableSet s) hμs 1) a • c) x = Set.indicator s (fun x => c) x ** dsimp only ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G x : α hx : ↑↑(indicatorConstLp 2 (_ : MeasurableSet s) hμs 1) x = Set.indicator s (fun x => 1) x ⊢ ↑↑(indicatorConstLp 2 (_ : MeasurableSet s) hμs 1) x • c = Set.indicator s (fun x => c) x ** rw [hx] ** case h α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹² : IsROrC 𝕜 inst✝¹¹ : NormedAddCommGroup F inst✝¹⁰ : NormedSpace 𝕜 F inst✝⁹ : NormedAddCommGroup F' inst✝⁸ : NormedSpace 𝕜 F' inst✝⁷ : NormedSpace ℝ F' inst✝⁶ : CompleteSpace F' inst✝⁵ : NormedAddCommGroup G inst✝⁴ : NormedAddCommGroup G' inst✝³ : NormedSpace ℝ G' inst✝² : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α s t : Set α inst✝¹ : NormedSpace ℝ G hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : G x : α hx : ↑↑(indicatorConstLp 2 (_ : MeasurableSet s) hμs 1) x = Set.indicator s (fun x => 1) x ⊢ Set.indicator s (fun x => 1) x • c = Set.indicator s (fun x => c) x ** by_cases hx_mem : x ∈ s <;> simp [hx_mem] ** Qed
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MeasureTheory.condexpL1Clm_smul ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹¹ : IsROrC 𝕜 inst✝¹⁰ : NormedAddCommGroup F inst✝⁹ : NormedSpace 𝕜 F inst✝⁸ : NormedAddCommGroup F' inst✝⁷ : NormedSpace 𝕜 F' inst✝⁶ : NormedSpace ℝ F' inst✝⁵ : CompleteSpace F' inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace ℝ G' inst✝¹ : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) f✝ g : α → F' s : Set α c : 𝕜 f : { x // x ∈ Lp F' 1 } ⊢ ↑(condexpL1Clm F' hm μ) (c • f) = c • ↑(condexpL1Clm F' hm μ) f ** refine' L1.setToL1_smul (dominatedFinMeasAdditive_condexpInd F' hm μ) _ c f ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹¹ : IsROrC 𝕜 inst✝¹⁰ : NormedAddCommGroup F inst✝⁹ : NormedSpace 𝕜 F inst✝⁸ : NormedAddCommGroup F' inst✝⁷ : NormedSpace 𝕜 F' inst✝⁶ : NormedSpace ℝ F' inst✝⁵ : CompleteSpace F' inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace ℝ G' inst✝¹ : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) f✝ g : α → F' s : Set α c : 𝕜 f : { x // x ∈ Lp F' 1 } ⊢ ∀ (c : 𝕜) (s : Set α) (x : F'), ↑(condexpInd F' hm μ s) (c • x) = c • ↑(condexpInd F' hm μ s) x ** exact fun c s x => condexpInd_smul' c x ** Qed
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MeasureTheory.condexpL1Clm_indicatorConst ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹¹ : IsROrC 𝕜 inst✝¹⁰ : NormedAddCommGroup F inst✝⁹ : NormedSpace 𝕜 F inst✝⁸ : NormedAddCommGroup F' inst✝⁷ : NormedSpace 𝕜 F' inst✝⁶ : NormedSpace ℝ F' inst✝⁵ : CompleteSpace F' inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace ℝ G' inst✝¹ : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) f g : α → F' s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : F' ⊢ ↑(condexpL1Clm F' hm μ) ↑(simpleFunc.indicatorConst 1 hs hμs x) = ↑(condexpInd F' hm μ s) x ** rw [Lp.simpleFunc.coe_indicatorConst] ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹¹ : IsROrC 𝕜 inst✝¹⁰ : NormedAddCommGroup F inst✝⁹ : NormedSpace 𝕜 F inst✝⁸ : NormedAddCommGroup F' inst✝⁷ : NormedSpace 𝕜 F' inst✝⁶ : NormedSpace ℝ F' inst✝⁵ : CompleteSpace F' inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace ℝ G' inst✝¹ : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) f g : α → F' s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ x : F' ⊢ ↑(condexpL1Clm F' hm μ) (indicatorConstLp 1 hs hμs x) = ↑(condexpInd F' hm μ s) x ** exact condexpL1Clm_indicatorConstLp hs hμs x ** Qed
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MeasureTheory.condexpL1_smul ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹¹ : IsROrC 𝕜 inst✝¹⁰ : NormedAddCommGroup F inst✝⁹ : NormedSpace 𝕜 F inst✝⁸ : NormedAddCommGroup F' inst✝⁷ : NormedSpace 𝕜 F' inst✝⁶ : NormedSpace ℝ F' inst✝⁵ : CompleteSpace F' inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace ℝ G' inst✝¹ : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) f✝ g : α → F' s : Set α c : 𝕜 f : α → F' ⊢ condexpL1 hm μ (c • f) = c • condexpL1 hm μ f ** refine' setToFun_smul _ _ c f ** α : Type u_1 β : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 𝕜 : Type u_7 p : ℝ≥0∞ inst✝¹¹ : IsROrC 𝕜 inst✝¹⁰ : NormedAddCommGroup F inst✝⁹ : NormedSpace 𝕜 F inst✝⁸ : NormedAddCommGroup F' inst✝⁷ : NormedSpace 𝕜 F' inst✝⁶ : NormedSpace ℝ F' inst✝⁵ : CompleteSpace F' inst✝⁴ : NormedAddCommGroup G inst✝³ : NormedAddCommGroup G' inst✝² : NormedSpace ℝ G' inst✝¹ : CompleteSpace G' m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) f✝ g : α → F' s : Set α c : 𝕜 f : α → F' ⊢ ∀ (c : 𝕜) (s : Set α) (x : F'), ↑(condexpInd F' hm μ s) (c • x) = c • ↑(condexpInd F' hm μ s) x ** exact fun c _ x => condexpInd_smul' c x ** Qed
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IsPiSystem.prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 C : Set (Set α) D : Set (Set β) hC : IsPiSystem C hD : IsPiSystem D ⊢ IsPiSystem (image2 (fun x x_1 => x ×ˢ x_1) C D) ** rintro _ ⟨s₁, t₁, hs₁, ht₁, rfl⟩ _ ⟨s₂, t₂, hs₂, ht₂, rfl⟩ hst ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 C : Set (Set α) D : Set (Set β) hC : IsPiSystem C hD : IsPiSystem D s₁ : Set α t₁ : Set β hs₁ : s₁ ∈ C ht₁ : t₁ ∈ D s₂ : Set α t₂ : Set β hs₂ : s₂ ∈ C ht₂ : t₂ ∈ D hst : Set.Nonempty ((fun x x_1 => x ×ˢ x_1) s₁ t₁ ∩ (fun x x_1 => x ×ˢ x_1) s₂ t₂) ⊢ (fun x x_1 => x ×ˢ x_1) s₁ t₁ ∩ (fun x x_1 => x ×ˢ x_1) s₂ t₂ ∈ image2 (fun x x_1 => x ×ˢ x_1) C D ** rw [prod_inter_prod] at hst ⊢ ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 C : Set (Set α) D : Set (Set β) hC : IsPiSystem C hD : IsPiSystem D s₁ : Set α t₁ : Set β hs₁ : s₁ ∈ C ht₁ : t₁ ∈ D s₂ : Set α t₂ : Set β hs₂ : s₂ ∈ C ht₂ : t₂ ∈ D hst : Set.Nonempty ((s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂)) ⊢ (s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂) ∈ image2 (fun x x_1 => x ×ˢ x_1) C D ** rw [prod_nonempty_iff] at hst ** case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 C : Set (Set α) D : Set (Set β) hC : IsPiSystem C hD : IsPiSystem D s₁ : Set α t₁ : Set β hs₁ : s₁ ∈ C ht₁ : t₁ ∈ D s₂ : Set α t₂ : Set β hs₂ : s₂ ∈ C ht₂ : t₂ ∈ D hst : Set.Nonempty (s₁ ∩ s₂) ∧ Set.Nonempty (t₁ ∩ t₂) ⊢ (s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂) ∈ image2 (fun x x_1 => x ×ˢ x_1) C D ** exact mem_image2_of_mem (hC _ hs₁ _ hs₂ hst.1) (hD _ ht₁ _ ht₂ hst.2) ** Qed
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generateFrom_prod_eq ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D ⊢ Prod.instMeasurableSpace = generateFrom (image2 (fun x x_1 => x ×ˢ x_1) C D) ** apply le_antisymm ** case a α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D ⊢ Prod.instMeasurableSpace ≤ generateFrom (image2 (fun x x_1 => x ×ˢ x_1) C D) ** refine' sup_le _ _ <;> rw [comap_generateFrom] <;> apply generateFrom_le <;>
rintro _ ⟨s, hs, rfl⟩ ** case a.refine'_1.h.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D s : Set α hs : s ∈ C ⊢ MeasurableSet (Prod.fst ⁻¹' s) ** rcases hD with ⟨t, h1t, h2t⟩ ** case a.refine'_1.h.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C s : Set α hs : s ∈ C t : ℕ → Set β h1t : ∀ (n : ℕ), t n ∈ D h2t : ⋃ n, t n = univ ⊢ MeasurableSet (Prod.fst ⁻¹' s) ** rw [← prod_univ, ← h2t, prod_iUnion] ** case a.refine'_1.h.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C s : Set α hs : s ∈ C t : ℕ → Set β h1t : ∀ (n : ℕ), t n ∈ D h2t : ⋃ n, t n = univ ⊢ MeasurableSet (⋃ i, s ×ˢ t i) ** apply MeasurableSet.iUnion ** case a.refine'_1.h.intro.intro.intro.intro.h α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C s : Set α hs : s ∈ C t : ℕ → Set β h1t : ∀ (n : ℕ), t n ∈ D h2t : ⋃ n, t n = univ ⊢ ∀ (b : ℕ), MeasurableSet (s ×ˢ t b) ** intro n ** case a.refine'_1.h.intro.intro.intro.intro.h α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C s : Set α hs : s ∈ C t : ℕ → Set β h1t : ∀ (n : ℕ), t n ∈ D h2t : ⋃ n, t n = univ n : ℕ ⊢ MeasurableSet (s ×ˢ t n) ** apply measurableSet_generateFrom ** case a.refine'_1.h.intro.intro.intro.intro.h.ht α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C s : Set α hs : s ∈ C t : ℕ → Set β h1t : ∀ (n : ℕ), t n ∈ D h2t : ⋃ n, t n = univ n : ℕ ⊢ s ×ˢ t n ∈ image2 (fun x x_1 => x ×ˢ x_1) C D ** exact ⟨s, t n, hs, h1t n, rfl⟩ ** case a.refine'_2.h.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D s : Set β hs : s ∈ D ⊢ MeasurableSet (Prod.snd ⁻¹' s) ** rcases hC with ⟨t, h1t, h2t⟩ ** case a.refine'_2.h.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hD : IsCountablySpanning D s : Set β hs : s ∈ D t : ℕ → Set α h1t : ∀ (n : ℕ), t n ∈ C h2t : ⋃ n, t n = univ ⊢ MeasurableSet (Prod.snd ⁻¹' s) ** rw [← univ_prod, ← h2t, iUnion_prod_const] ** case a.refine'_2.h.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hD : IsCountablySpanning D s : Set β hs : s ∈ D t : ℕ → Set α h1t : ∀ (n : ℕ), t n ∈ C h2t : ⋃ n, t n = univ ⊢ MeasurableSet (⋃ i, t i ×ˢ s) ** apply MeasurableSet.iUnion ** case a.refine'_2.h.intro.intro.intro.intro.h α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hD : IsCountablySpanning D s : Set β hs : s ∈ D t : ℕ → Set α h1t : ∀ (n : ℕ), t n ∈ C h2t : ⋃ n, t n = univ ⊢ ∀ (b : ℕ), MeasurableSet (t b ×ˢ s) ** rintro n ** case a.refine'_2.h.intro.intro.intro.intro.h α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hD : IsCountablySpanning D s : Set β hs : s ∈ D t : ℕ → Set α h1t : ∀ (n : ℕ), t n ∈ C h2t : ⋃ n, t n = univ n : ℕ ⊢ MeasurableSet (t n ×ˢ s) ** apply measurableSet_generateFrom ** case a.refine'_2.h.intro.intro.intro.intro.h.ht α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hD : IsCountablySpanning D s : Set β hs : s ∈ D t : ℕ → Set α h1t : ∀ (n : ℕ), t n ∈ C h2t : ⋃ n, t n = univ n : ℕ ⊢ t n ×ˢ s ∈ image2 (fun x x_1 => x ×ˢ x_1) C D ** exact mem_image2_of_mem (h1t n) hs ** case a α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D ⊢ generateFrom (image2 (fun x x_1 => x ×ˢ x_1) C D) ≤ Prod.instMeasurableSpace ** apply generateFrom_le ** case a.h α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D ⊢ ∀ (t : Set (α × β)), t ∈ image2 (fun x x_1 => x ×ˢ x_1) C D → MeasurableSet t ** rintro _ ⟨s, t, hs, ht, rfl⟩ ** case a.h.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D s : Set α t : Set β hs : s ∈ C ht : t ∈ D ⊢ MeasurableSet ((fun x x_1 => x ×ˢ x_1) s t) ** dsimp only ** case a.h.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D s : Set α t : Set β hs : s ∈ C ht : t ∈ D ⊢ MeasurableSet (s ×ˢ t) ** rw [prod_eq] ** case a.h.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D s : Set α t : Set β hs : s ∈ C ht : t ∈ D ⊢ MeasurableSet (Prod.fst ⁻¹' s ∩ Prod.snd ⁻¹' t) ** apply (measurable_fst _).inter (measurable_snd _) ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D s : Set α t : Set β hs : s ∈ C ht : t ∈ D ⊢ MeasurableSet s ** exact measurableSet_generateFrom hs ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 C : Set (Set α) D : Set (Set β) hC : IsCountablySpanning C hD : IsCountablySpanning D s : Set α t : Set β hs : s ∈ C ht : t ∈ D ⊢ MeasurableSet t ** exact measurableSet_generateFrom ht ** Qed
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measurable_measure_prod_mk_left_finite ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s) ** refine' induction_on_inter (C := fun s => Measurable fun x => ν (Prod.mk x ⁻¹' s))
generateFrom_prod.symm isPiSystem_prod _ _ _ _ hs ** case refine'_1 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s ⊢ (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) ∅ ** simp [measurable_zero, const_def] ** case refine'_2 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s ⊢ ∀ (t : Set (α × β)), t ∈ image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t} → (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) t ** rintro _ ⟨s, t, hs, _, rfl⟩ ** case refine'_2.intro.intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s✝ : Set (α × β) hs✝ : MeasurableSet s✝ s : Set α t : Set β hs : s ∈ {s | MeasurableSet s} left✝ : t ∈ {t | MeasurableSet t} ⊢ Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' (fun x x_1 => x ×ˢ x_1) s t) ** simp only [mk_preimage_prod_right_eq_if, measure_if] ** case refine'_2.intro.intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s✝ : Set (α × β) hs✝ : MeasurableSet s✝ s : Set α t : Set β hs : s ∈ {s | MeasurableSet s} left✝ : t ∈ {t | MeasurableSet t} ⊢ Measurable fun x => indicator s (fun x => ↑↑ν t) x ** exact measurable_const.indicator hs ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s ⊢ ∀ (t : Set (α × β)), MeasurableSet t → (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) t → (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) tᶜ ** intro t ht h2t ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s t : Set (α × β) ht : MeasurableSet t h2t : Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' t) ⊢ Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' tᶜ) ** simp_rw [preimage_compl, measure_compl (measurable_prod_mk_left ht) (measure_ne_top ν _)] ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s t : Set (α × β) ht : MeasurableSet t h2t : Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' t) ⊢ Measurable fun x => ↑↑ν univ - ↑↑ν (Prod.mk x ⁻¹' t) ** exact h2t.const_sub _ ** case refine'_4 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s ⊢ ∀ (f : ℕ → Set (α × β)), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) (f i)) → (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) (⋃ i, f i) ** intro f h1f h2f h3f ** case refine'_4 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s f : ℕ → Set (α × β) h1f : Pairwise (Disjoint on f) h2f : ∀ (i : ℕ), MeasurableSet (f i) h3f : ∀ (i : ℕ), (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) (f i) ⊢ Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' ⋃ i, f i) ** simp_rw [preimage_iUnion] ** case refine'_4 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s f : ℕ → Set (α × β) h1f : Pairwise (Disjoint on f) h2f : ∀ (i : ℕ), MeasurableSet (f i) h3f : ∀ (i : ℕ), (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) (f i) ⊢ Measurable fun x => ↑↑ν (⋃ i, Prod.mk x ⁻¹' f i) ** have : ∀ b, ν (⋃ i, Prod.mk b ⁻¹' f i) = ∑' i, ν (Prod.mk b ⁻¹' f i) := fun b =>
measure_iUnion (fun i j hij => Disjoint.preimage _ (h1f hij)) fun i =>
measurable_prod_mk_left (h2f i) ** case refine'_4 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s f : ℕ → Set (α × β) h1f : Pairwise (Disjoint on f) h2f : ∀ (i : ℕ), MeasurableSet (f i) h3f : ∀ (i : ℕ), (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) (f i) this : ∀ (b : α), ↑↑ν (⋃ i, Prod.mk b ⁻¹' f i) = ∑' (i : ℕ), ↑↑ν (Prod.mk b ⁻¹' f i) ⊢ Measurable fun x => ↑↑ν (⋃ i, Prod.mk x ⁻¹' f i) ** simp_rw [this] ** case refine'_4 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : IsFiniteMeasure ν s : Set (α × β) hs : MeasurableSet s f : ℕ → Set (α × β) h1f : Pairwise (Disjoint on f) h2f : ∀ (i : ℕ), MeasurableSet (f i) h3f : ∀ (i : ℕ), (fun s => Measurable fun x => ↑↑ν (Prod.mk x ⁻¹' s)) (f i) this : ∀ (b : α), ↑↑ν (⋃ i, Prod.mk b ⁻¹' f i) = ∑' (i : ℕ), ↑↑ν (Prod.mk b ⁻¹' f i) ⊢ Measurable fun x => ∑' (i : ℕ), ↑↑ν (Prod.mk x ⁻¹' f i) ** apply Measurable.ennreal_tsum h3f ** Qed
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Measurable.map_prod_mk_left ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ⊢ Measurable fun x => map (Prod.mk x) ν ** apply measurable_of_measurable_coe ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ⊢ ∀ (s : Set (α × β)), MeasurableSet s → Measurable fun b => ↑↑(map (Prod.mk b) ν) s ** intro s hs ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(map (Prod.mk b) ν) s ** simp_rw [map_apply measurable_prod_mk_left hs] ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun b => ↑↑ν (Prod.mk b ⁻¹' s) ** exact measurable_measure_prod_mk_left hs ** Qed
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Measurable.map_prod_mk_right ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E μ : Measure α inst✝ : SigmaFinite μ ⊢ Measurable fun y => map (fun x => (x, y)) μ ** apply measurable_of_measurable_coe ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E μ : Measure α inst✝ : SigmaFinite μ ⊢ ∀ (s : Set (α × β)), MeasurableSet s → Measurable fun b => ↑↑(map (fun x => (x, b)) μ) s ** intro s hs ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E μ : Measure α inst✝ : SigmaFinite μ s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(map (fun x => (x, b)) μ) s ** simp_rw [map_apply measurable_prod_mk_right hs] ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E μ : Measure α inst✝ : SigmaFinite μ s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun b => ↑↑μ ((fun x => (x, b)) ⁻¹' s) ** exact measurable_measure_prod_mk_right hs ** Qed
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MeasurableEmbedding.prod_mk ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f ⊢ MeasurableEmbedding fun x => (g x.1, f x.2) ** have h_inj : Function.Injective fun x : γ × α => (g x.fst, f x.snd) := by
intro x y hxy
rw [← @Prod.mk.eta _ _ x, ← @Prod.mk.eta _ _ y]
simp only [Prod.mk.inj_iff] at hxy ⊢
exact ⟨hg.injective hxy.1, hf.injective hxy.2⟩ ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) ⊢ MeasurableEmbedding fun x => (g x.1, f x.2) ** refine' ⟨h_inj, _, _⟩ ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f ⊢ Injective fun x => (g x.1, f x.2) ** intro x y hxy ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f x y : γ × α hxy : (fun x => (g x.1, f x.2)) x = (fun x => (g x.1, f x.2)) y ⊢ x = y ** rw [← @Prod.mk.eta _ _ x, ← @Prod.mk.eta _ _ y] ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f x y : γ × α hxy : (fun x => (g x.1, f x.2)) x = (fun x => (g x.1, f x.2)) y ⊢ (x.1, x.2) = (y.1, y.2) ** simp only [Prod.mk.inj_iff] at hxy ⊢ ** α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f x y : γ × α hxy : g x.1 = g y.1 ∧ f x.2 = f y.2 ⊢ x.1 = y.1 ∧ x.2 = y.2 ** exact ⟨hg.injective hxy.1, hf.injective hxy.2⟩ ** case refine'_1 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) ⊢ Measurable fun x => (g x.1, f x.2) ** exact (hg.measurable.comp measurable_fst).prod_mk (hf.measurable.comp measurable_snd) ** case refine'_2 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) ⊢ ∀ ⦃s : Set (γ × α)⦄, MeasurableSet s → MeasurableSet ((fun x => (g x.1, f x.2)) '' s) ** refine' fun s hs =>
@MeasurableSpace.induction_on_inter _
(fun s => MeasurableSet ((fun x : γ × α => (g x.fst, f x.snd)) '' s)) _ _
generateFrom_prod.symm isPiSystem_prod _ _ _ _ _ hs ** case refine'_2.refine'_1 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s ⊢ (fun s => MeasurableSet ((fun x => (g x.1, f x.2)) '' s)) ∅ ** simp only [Set.image_empty, MeasurableSet.empty] ** case refine'_2.refine'_2 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s ⊢ ∀ (t : Set (γ × α)), t ∈ image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t} → (fun s => MeasurableSet ((fun x => (g x.1, f x.2)) '' s)) t ** rintro t ⟨t₁, t₂, ht₁, ht₂, rfl⟩ ** case refine'_2.refine'_2.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s t₁ : Set γ t₂ : Set α ht₁ : t₁ ∈ {s | MeasurableSet s} ht₂ : t₂ ∈ {t | MeasurableSet t} ⊢ MeasurableSet ((fun x => (g x.1, f x.2)) '' (fun x x_1 => x ×ˢ x_1) t₁ t₂) ** rw [← Set.prod_image_image_eq] ** case refine'_2.refine'_2.intro.intro.intro.intro α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s t₁ : Set γ t₂ : Set α ht₁ : t₁ ∈ {s | MeasurableSet s} ht₂ : t₂ ∈ {t | MeasurableSet t} ⊢ MeasurableSet ((g '' t₁) ×ˢ (f '' t₂)) ** exact (hg.measurableSet_image.mpr ht₁).prod (hf.measurableSet_image.mpr ht₂) ** case refine'_2.refine'_3 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s ⊢ ∀ (t : Set (γ × α)), MeasurableSet t → (fun s => MeasurableSet ((fun x => (g x.1, f x.2)) '' s)) t → (fun s => MeasurableSet ((fun x => (g x.1, f x.2)) '' s)) tᶜ ** intro t _ ht_m ** case refine'_2.refine'_3 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s t : Set (γ × α) a✝ : MeasurableSet t ht_m : MeasurableSet ((fun x => (g x.1, f x.2)) '' t) ⊢ MeasurableSet ((fun x => (g x.1, f x.2)) '' tᶜ) ** rw [← Set.range_diff_image h_inj, ← Set.prod_range_range_eq] ** case refine'_2.refine'_3 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s t : Set (γ × α) a✝ : MeasurableSet t ht_m : MeasurableSet ((fun x => (g x.1, f x.2)) '' t) ⊢ MeasurableSet (range g ×ˢ range f \ (fun x => (g x.1, f x.2)) '' t) ** exact
MeasurableSet.diff (MeasurableSet.prod hg.measurableSet_range hf.measurableSet_range) ht_m ** case refine'_2.refine'_4 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g : γ → δ hg : MeasurableEmbedding g hf : MeasurableEmbedding f h_inj : Injective fun x => (g x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s ⊢ ∀ (f_1 : ℕ → Set (γ × α)), Pairwise (Disjoint on f_1) → (∀ (i : ℕ), MeasurableSet (f_1 i)) → (∀ (i : ℕ), (fun s => MeasurableSet ((fun x => (g x.1, f x.2)) '' s)) (f_1 i)) → (fun s => MeasurableSet ((fun x => (g x.1, f x.2)) '' s)) (⋃ i, f_1 i) ** intro g _ _ hg ** case refine'_2.refine'_4 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g✝ : γ → δ hg✝ : MeasurableEmbedding g✝ hf : MeasurableEmbedding f h_inj : Injective fun x => (g✝ x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s g : ℕ → Set (γ × α) a✝¹ : Pairwise (Disjoint on g) a✝ : ∀ (i : ℕ), MeasurableSet (g i) hg : ∀ (i : ℕ), (fun s => MeasurableSet ((fun x => (g✝ x.1, f x.2)) '' s)) (g i) ⊢ MeasurableSet ((fun x => (g✝ x.1, f x.2)) '' ⋃ i, g i) ** simp_rw [Set.image_iUnion] ** case refine'_2.refine'_4 α✝ : Type u_1 α' : Type u_2 β✝ : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁵ : MeasurableSpace α✝ inst✝⁴ : MeasurableSpace α' inst✝³ : MeasurableSpace β✝ inst✝² : MeasurableSpace β' inst✝¹ : MeasurableSpace γ✝ μ μ' : Measure α✝ ν ν' : Measure β✝ τ : Measure γ✝ inst✝ : NormedAddCommGroup E α : Type u_7 β : Type u_8 γ : Type u_9 δ : Type u_10 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ mδ : MeasurableSpace δ f : α → β g✝ : γ → δ hg✝ : MeasurableEmbedding g✝ hf : MeasurableEmbedding f h_inj : Injective fun x => (g✝ x.1, f x.2) s : Set (γ × α) hs : MeasurableSet s g : ℕ → Set (γ × α) a✝¹ : Pairwise (Disjoint on g) a✝ : ∀ (i : ℕ), MeasurableSet (g i) hg : ∀ (i : ℕ), (fun s => MeasurableSet ((fun x => (g✝ x.1, f x.2)) '' s)) (g i) ⊢ MeasurableSet (⋃ i, (fun x => (g✝ x.1, f x.2)) '' g i) ** exact MeasurableSet.iUnion hg ** Qed
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Measurable.lintegral_prod_right' ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ⊢ ∀ {f : α × β → ℝ≥0∞}, Measurable f → Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν ** have m := @measurable_prod_mk_left ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) ⊢ ∀ {f : α × β → ℝ≥0∞}, Measurable f → Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν ** refine' Measurable.ennreal_induction (P := fun f => Measurable fun (x : α) => ∫⁻ y, f (x, y) ∂ν)
_ _ _ ** case refine'_1 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) ⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)⦄, MeasurableSet s → (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (Set.indicator s fun x => c) ** intro c s hs ** case refine'_1 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) c : ℝ≥0∞ s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator s (fun x => c) (x, y) ∂ν ** simp only [← indicator_comp_right] ** case refine'_1 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) c : ℝ≥0∞ s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator (Prod.mk x ⁻¹' s) ((fun x => c) ∘ Prod.mk x) y ∂ν ** suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator _ (m hs)] ** case refine'_1 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) c : ℝ≥0∞ s : Set (α × β) hs : MeasurableSet s ⊢ Measurable fun x => c * ↑↑ν (Prod.mk x ⁻¹' s) ** exact (measurable_measure_prod_mk_left hs).const_mul _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) c : ℝ≥0∞ s : Set (α × β) hs : MeasurableSet s this : Measurable fun x => c * ↑↑ν (Prod.mk x ⁻¹' s) ⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator (Prod.mk x ⁻¹' s) ((fun x => c) ∘ Prod.mk x) y ∂ν ** simpa [lintegral_indicator _ (m hs)] ** case refine'_2 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) ⊢ ∀ ⦃f g : α × β → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) f → (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) g → (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f + g) ** rintro f g - hf - h2f h2g ** case refine'_2 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) f g : α × β → ℝ≥0∞ hf : Measurable f h2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν h2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν ⊢ Measurable fun x => ∫⁻ (y : β), (f + g) (x, y) ∂ν ** simp only [Pi.add_apply] ** case refine'_2 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type ?u.13744} {β : Type ?u.13745} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) f g : α × β → ℝ≥0∞ hf : Measurable f h2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν h2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν ⊢ Measurable fun x => ∫⁻ (y : β), f (x, y) + g (x, y) ∂ν ** conv => enter [1, x]; erw [lintegral_add_left (hf.comp m)] ** case refine'_2 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type u_1} {β : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) f g : α × β → ℝ≥0∞ hf : Measurable f h2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν h2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν ⊢ Measurable fun x => ∫⁻ (a : β), (f ∘ Prod.mk x) a ∂ν + ∫⁻ (a : β), g (x, a) ∂ν ** exact h2f.add h2g ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type u_1} {β : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) ⊢ ∀ ⦃f : ℕ → α × β → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)) → (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) fun x => ⨆ n, f n x ** intro f hf h2f h3f ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type u_1} {β : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) f : ℕ → α × β → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h2f : Monotone f h3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n) ⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ n, f n x) (x, y) ∂ν ** have := measurable_iSup h3f ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type u_1} {β : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) f : ℕ → α × β → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h2f : Monotone f h3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n) this : Measurable fun b => ⨆ i, ∫⁻ (y : β), f i (b, y) ∂ν ⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ n, f n x) (x, y) ∂ν ** have : ∀ x, Monotone fun n y => f n (x, y) := fun x i j hij y => h2f hij (x, y) ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type u_1} {β : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) f : ℕ → α × β → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h2f : Monotone f h3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n) this✝ : Measurable fun b => ⨆ i, ∫⁻ (y : β), f i (b, y) ∂ν this : ∀ (x : α), Monotone fun n y => f n (x, y) ⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ n, f n x) (x, y) ∂ν ** conv => enter [1, x]; erw [lintegral_iSup (fun n => (hf n).comp m) (this x)] ** case refine'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν m : ∀ {α : Type u_1} {β : Type u_3} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x) f : ℕ → α × β → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h2f : Monotone f h3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n) this✝ : Measurable fun b => ⨆ i, ∫⁻ (y : β), f i (b, y) ∂ν this : ∀ (x : α), Monotone fun n y => f n (x, y) ⊢ Measurable fun x => ⨆ n, ∫⁻ (a : β), (f n ∘ Prod.mk x) a ∂ν ** assumption ** Qed
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MeasureTheory.Measure.prod_prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ⊢ ↑↑(Measure.prod μ ν) (s ×ˢ t) = ↑↑μ s * ↑↑ν t ** apply le_antisymm ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ⊢ ↑↑(Measure.prod μ ν) (s ×ˢ t) ≤ ↑↑μ s * ↑↑ν t ** set ST := toMeasurable μ s ×ˢ toMeasurable ν t ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable μ s ×ˢ toMeasurable ν t ⊢ ↑↑(Measure.prod μ ν) (s ×ˢ t) ≤ ↑↑μ s * ↑↑ν t ** have hSTm : MeasurableSet ST :=
(measurableSet_toMeasurable _ _).prod (measurableSet_toMeasurable _ _) ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable μ s ×ˢ toMeasurable ν t hSTm : MeasurableSet ST ⊢ ↑↑(Measure.prod μ ν) (s ×ˢ t) ≤ ↑↑μ s * ↑↑ν t ** calc
μ.prod ν (s ×ˢ t) ≤ μ.prod ν ST :=
measure_mono <| Set.prod_mono (subset_toMeasurable _ _) (subset_toMeasurable _ _)
_ = μ (toMeasurable μ s) * ν (toMeasurable ν t) := by
rw [prod_apply hSTm]
simp_rw [mk_preimage_prod_right_eq_if, measure_if,
lintegral_indicator _ (measurableSet_toMeasurable _ _), lintegral_const,
restrict_apply_univ, mul_comm]
_ = μ s * ν t := by rw [measure_toMeasurable, measure_toMeasurable] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable μ s ×ˢ toMeasurable ν t hSTm : MeasurableSet ST ⊢ ↑↑(Measure.prod μ ν) ST = ↑↑μ (toMeasurable μ s) * ↑↑ν (toMeasurable ν t) ** rw [prod_apply hSTm] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable μ s ×ˢ toMeasurable ν t hSTm : MeasurableSet ST ⊢ ∫⁻ (x : α), ↑↑ν (Prod.mk x ⁻¹' ST) ∂μ = ↑↑μ (toMeasurable μ s) * ↑↑ν (toMeasurable ν t) ** simp_rw [mk_preimage_prod_right_eq_if, measure_if,
lintegral_indicator _ (measurableSet_toMeasurable _ _), lintegral_const,
restrict_apply_univ, mul_comm] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable μ s ×ˢ toMeasurable ν t hSTm : MeasurableSet ST ⊢ ↑↑μ (toMeasurable μ s) * ↑↑ν (toMeasurable ν t) = ↑↑μ s * ↑↑ν t ** rw [measure_toMeasurable, measure_toMeasurable] ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** set ST := toMeasurable (μ.prod ν) (s ×ˢ t) ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** have hSTm : MeasurableSet ST := measurableSet_toMeasurable _ _ ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) hSTm : MeasurableSet ST ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** have hST : s ×ˢ t ⊆ ST := subset_toMeasurable _ _ ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) hSTm : MeasurableSet ST hST : s ×ˢ t ⊆ ST ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** set f : α → ℝ≥0∞ := fun x => ν (Prod.mk x ⁻¹' ST) ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) hSTm : MeasurableSet ST hST : s ×ˢ t ⊆ ST f : α → ℝ≥0∞ := fun x => ↑↑ν (Prod.mk x ⁻¹' ST) ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** have hfm : Measurable f := measurable_measure_prod_mk_left hSTm ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) hSTm : MeasurableSet ST hST : s ×ˢ t ⊆ ST f : α → ℝ≥0∞ := fun x => ↑↑ν (Prod.mk x ⁻¹' ST) hfm : Measurable f ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** set s' : Set α := { x | ν t ≤ f x } ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) hSTm : MeasurableSet ST hST : s ×ˢ t ⊆ ST f : α → ℝ≥0∞ := fun x => ↑↑ν (Prod.mk x ⁻¹' ST) hfm : Measurable f s' : Set α := {x | ↑↑ν t ≤ f x} ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** have hss' : s ⊆ s' := fun x hx => measure_mono fun y hy => hST <| mk_mem_prod hx hy ** case a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) hSTm : MeasurableSet ST hST : s ×ˢ t ⊆ ST f : α → ℝ≥0∞ := fun x => ↑↑ν (Prod.mk x ⁻¹' ST) hfm : Measurable f s' : Set α := {x | ↑↑ν t ≤ f x} hss' : s ⊆ s' ⊢ ↑↑μ s * ↑↑ν t ≤ ↑↑(Measure.prod μ ν) (s ×ˢ t) ** calc
μ s * ν t ≤ μ s' * ν t := mul_le_mul_right' (measure_mono hss') _
_ = ∫⁻ _ in s', ν t ∂μ := by rw [set_lintegral_const, mul_comm]
_ ≤ ∫⁻ x in s', f x ∂μ := (set_lintegral_mono measurable_const hfm fun x => id)
_ ≤ ∫⁻ x, f x ∂μ := (lintegral_mono' restrict_le_self le_rfl)
_ = μ.prod ν ST := (prod_apply hSTm).symm
_ = μ.prod ν (s ×ˢ t) := measure_toMeasurable _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α t : Set β ST : Set (α × β) := toMeasurable (Measure.prod μ ν) (s ×ˢ t) hSTm : MeasurableSet ST hST : s ×ˢ t ⊆ ST f : α → ℝ≥0∞ := fun x => ↑↑ν (Prod.mk x ⁻¹' ST) hfm : Measurable f s' : Set α := {x | ↑↑ν t ≤ f x} hss' : s ⊆ s' ⊢ ↑↑μ s' * ↑↑ν t = ∫⁻ (x : α) in s', ↑↑ν t ∂μ ** rw [set_lintegral_const, mul_comm] ** Qed
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MeasureTheory.Measure.ae_measure_lt_top ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, ↑↑ν (Prod.mk x ⁻¹' s) < ⊤ ** rw [prod_apply hs] at h2s ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ∫⁻ (x : α), ↑↑ν (Prod.mk x ⁻¹' s) ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, ↑↑ν (Prod.mk x ⁻¹' s) < ⊤ ** refine' ae_lt_top (measurable_measure_prod_mk_left hs) h2s ** Qed
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MeasureTheory.Measure.measure_prod_null ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s ⊢ ↑↑(Measure.prod μ ν) s = 0 ↔ (fun x => ↑↑ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0 ** rw [prod_apply hs, lintegral_eq_zero_iff (measurable_measure_prod_mk_left hs)] ** Qed
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MeasureTheory.Measure.measure_ae_null_of_prod_null ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) h : ↑↑(Measure.prod μ ν) s = 0 ⊢ (fun x => ↑↑ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0 ** obtain ⟨t, hst, mt, ht⟩ := exists_measurable_superset_of_null h ** case intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) h : ↑↑(Measure.prod μ ν) s = 0 t : Set (α × β) hst : s ⊆ t mt : MeasurableSet t ht : ↑↑(Measure.prod μ ν) t = 0 ⊢ (fun x => ↑↑ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0 ** rw [measure_prod_null mt] at ht ** case intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) h : ↑↑(Measure.prod μ ν) s = 0 t : Set (α × β) hst : s ⊆ t mt : MeasurableSet t ht : (fun x => ↑↑ν (Prod.mk x ⁻¹' t)) =ᶠ[ae μ] 0 ⊢ (fun x => ↑↑ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0 ** rw [eventuallyLE_antisymm_iff] ** case intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) h : ↑↑(Measure.prod μ ν) s = 0 t : Set (α × β) hst : s ⊆ t mt : MeasurableSet t ht : (fun x => ↑↑ν (Prod.mk x ⁻¹' t)) =ᶠ[ae μ] 0 ⊢ (fun x => ↑↑ν (Prod.mk x ⁻¹' s)) ≤ᶠ[ae μ] 0 ∧ 0 ≤ᶠ[ae μ] fun x => ↑↑ν (Prod.mk x ⁻¹' s) ** exact
⟨EventuallyLE.trans_eq (eventually_of_forall fun x => (measure_mono (preimage_mono hst) : _))
ht,
eventually_of_forall fun x => zero_le _⟩ ** Qed
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MeasureTheory.Measure.AbsolutelyContinuous.prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite ν' h1 : μ ≪ μ' h2 : ν ≪ ν' ⊢ Measure.prod μ ν ≪ Measure.prod μ' ν' ** refine' AbsolutelyContinuous.mk fun s hs h2s => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite ν' h1 : μ ≪ μ' h2 : ν ≪ ν' s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ' ν') s = 0 ⊢ ↑↑(Measure.prod μ ν) s = 0 ** rw [measure_prod_null hs] at h2s ⊢ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite ν' h1 : μ ≪ μ' h2 : ν ≪ ν' s : Set (α × β) hs : MeasurableSet s h2s : (fun x => ↑↑ν' (Prod.mk x ⁻¹' s)) =ᶠ[ae μ'] 0 ⊢ (fun x => ↑↑ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0 ** exact (h2s.filter_mono h1.ae_le).mono fun _ h => h2 h ** Qed
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MeasureTheory.Measure.quasiMeasurePreserving_fst ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ⊢ QuasiMeasurePreserving Prod.fst ** refine' ⟨measurable_fst, AbsolutelyContinuous.mk fun s hs h2s => _⟩ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set α hs : MeasurableSet s h2s : ↑↑μ s = 0 ⊢ ↑↑(map Prod.fst (Measure.prod μ ν)) s = 0 ** rw [map_apply measurable_fst hs, ← prod_univ, prod_prod, h2s, zero_mul] ** Qed
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MeasureTheory.Measure.quasiMeasurePreserving_snd ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ⊢ QuasiMeasurePreserving Prod.snd ** refine' ⟨measurable_snd, AbsolutelyContinuous.mk fun s hs h2s => _⟩ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set β hs : MeasurableSet s h2s : ↑↑ν s = 0 ⊢ ↑↑(map Prod.snd (Measure.prod μ ν)) s = 0 ** rw [map_apply measurable_snd hs, ← univ_prod, prod_prod, h2s, mul_zero] ** Qed
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MeasureTheory.Measure.prod_eq_generateFrom ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν✝ inst✝ : SigmaFinite μ✝ μ : Measure α ν : Measure β C : Set (Set α) D : Set (Set β) hC : generateFrom C = inst✝⁷ hD : generateFrom D = inst✝⁵ h2C : IsPiSystem C h2D : IsPiSystem D h3C : FiniteSpanningSetsIn μ C h3D : FiniteSpanningSetsIn ν D μν : Measure (α × β) h₁ : ∀ (s : Set α), s ∈ C → ∀ (t : Set β), t ∈ D → ↑↑μν (s ×ˢ t) = ↑↑μ s * ↑↑ν t ⊢ Measure.prod μ ν = μν ** refine'
(h3C.prod h3D).ext
(generateFrom_eq_prod hC hD h3C.isCountablySpanning h3D.isCountablySpanning).symm
(h2C.prod h2D) _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν✝ inst✝ : SigmaFinite μ✝ μ : Measure α ν : Measure β C : Set (Set α) D : Set (Set β) hC : generateFrom C = inst✝⁷ hD : generateFrom D = inst✝⁵ h2C : IsPiSystem C h2D : IsPiSystem D h3C : FiniteSpanningSetsIn μ C h3D : FiniteSpanningSetsIn ν D μν : Measure (α × β) h₁ : ∀ (s : Set α), s ∈ C → ∀ (t : Set β), t ∈ D → ↑↑μν (s ×ˢ t) = ↑↑μ s * ↑↑ν t ⊢ ∀ (s : Set (α × β)), s ∈ image2 (fun x x_1 => x ×ˢ x_1) C D → ↑↑(Measure.prod μ ν) s = ↑↑μν s ** rintro _ ⟨s, t, hs, ht, rfl⟩ ** case intro.intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν✝ inst✝ : SigmaFinite μ✝ μ : Measure α ν : Measure β C : Set (Set α) D : Set (Set β) hC : generateFrom C = inst✝⁷ hD : generateFrom D = inst✝⁵ h2C : IsPiSystem C h2D : IsPiSystem D h3C : FiniteSpanningSetsIn μ C h3D : FiniteSpanningSetsIn ν D μν : Measure (α × β) h₁ : ∀ (s : Set α), s ∈ C → ∀ (t : Set β), t ∈ D → ↑↑μν (s ×ˢ t) = ↑↑μ s * ↑↑ν t s : Set α t : Set β hs : s ∈ C ht : t ∈ D ⊢ ↑↑(Measure.prod μ ν) ((fun x x_1 => x ×ˢ x_1) s t) = ↑↑μν ((fun x x_1 => x ×ˢ x_1) s t) ** haveI := h3D.sigmaFinite ** case intro.intro.intro.intro α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν✝ inst✝ : SigmaFinite μ✝ μ : Measure α ν : Measure β C : Set (Set α) D : Set (Set β) hC : generateFrom C = inst✝⁷ hD : generateFrom D = inst✝⁵ h2C : IsPiSystem C h2D : IsPiSystem D h3C : FiniteSpanningSetsIn μ C h3D : FiniteSpanningSetsIn ν D μν : Measure (α × β) h₁ : ∀ (s : Set α), s ∈ C → ∀ (t : Set β), t ∈ D → ↑↑μν (s ×ˢ t) = ↑↑μ s * ↑↑ν t s : Set α t : Set β hs : s ∈ C ht : t ∈ D this : SigmaFinite ν ⊢ ↑↑(Measure.prod μ ν) ((fun x x_1 => x ×ˢ x_1) s t) = ↑↑μν ((fun x x_1 => x ×ˢ x_1) s t) ** rw [h₁ s hs t ht, prod_prod] ** Qed
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MeasureTheory.Measure.prod_swap ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ ⊢ map Prod.swap (Measure.prod μ ν) = Measure.prod ν μ ** refine' (prod_eq _).symm ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ ⊢ ∀ (s : Set β) (t : Set α), MeasurableSet s → MeasurableSet t → ↑↑(map Prod.swap (Measure.prod μ ν)) (s ×ˢ t) = ↑↑ν s * ↑↑μ t ** intro s t hs ht ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set β t : Set α hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑↑(map Prod.swap (Measure.prod μ ν)) (s ×ˢ t) = ↑↑ν s * ↑↑μ t ** simp_rw [map_apply measurable_swap (hs.prod ht), preimage_swap_prod, prod_prod, mul_comm] ** Qed
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MeasureTheory.Measure.prod_apply_symm ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set (α × β) hs : MeasurableSet s ⊢ ↑↑(Measure.prod μ ν) s = ∫⁻ (y : β), ↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂ν ** rw [← prod_swap, map_apply measurable_swap hs, prod_apply (measurable_swap hs)] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set (α × β) hs : MeasurableSet s ⊢ ∫⁻ (x : β), ↑↑μ (Prod.mk x ⁻¹' (Prod.swap ⁻¹' s)) ∂ν = ∫⁻ (y : β), ↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂ν ** rfl ** Qed
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MeasureTheory.Measure.prod_restrict ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set α t : Set β ⊢ Measure.prod (restrict μ s) (restrict ν t) = restrict (Measure.prod μ ν) (s ×ˢ t) ** refine' prod_eq fun s' t' hs' ht' => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set α t : Set β s' : Set α t' : Set β hs' : MeasurableSet s' ht' : MeasurableSet t' ⊢ ↑↑(restrict (Measure.prod μ ν) (s ×ˢ t)) (s' ×ˢ t') = ↑↑(restrict μ s) s' * ↑↑(restrict ν t) t' ** rw [restrict_apply (hs'.prod ht'), prod_inter_prod, prod_prod, restrict_apply hs',
restrict_apply ht'] ** Qed
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MeasureTheory.Measure.restrict_prod_eq_prod_univ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set α ⊢ Measure.prod (restrict μ s) ν = restrict (Measure.prod μ ν) (s ×ˢ univ) ** have : ν = ν.restrict Set.univ := Measure.restrict_univ.symm ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ s : Set α this : ν = restrict ν univ ⊢ Measure.prod (restrict μ s) ν = restrict (Measure.prod μ ν) (s ×ˢ univ) ** rw [this, Measure.prod_restrict, ← this] ** Qed
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MeasureTheory.Measure.prod_dirac ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ y : β ⊢ Measure.prod μ (dirac y) = map (fun x => (x, y)) μ ** refine' prod_eq fun s t hs ht => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ y : β s : Set α t : Set β hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑↑(map (fun x => (x, y)) μ) (s ×ˢ t) = ↑↑μ s * ↑↑(dirac y) t ** simp_rw [map_apply measurable_prod_mk_right (hs.prod ht), mk_preimage_prod_left_eq_if, measure_if,
dirac_apply' _ ht, ← indicator_mul_right _ fun _ => μ s, Pi.one_apply, mul_one] ** Qed
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MeasureTheory.Measure.dirac_prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ x : α ⊢ Measure.prod (dirac x) ν = map (Prod.mk x) ν ** refine' prod_eq fun s t hs ht => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ x : α s : Set α t : Set β hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑↑(map (Prod.mk x) ν) (s ×ˢ t) = ↑↑(dirac x) s * ↑↑ν t ** simp_rw [map_apply measurable_prod_mk_left (hs.prod ht), mk_preimage_prod_right_eq_if, measure_if,
dirac_apply' _ hs, ← indicator_mul_left _ _ fun _ => ν t, Pi.one_apply, one_mul] ** Qed
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MeasureTheory.Measure.prod_sum ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁹ : MeasurableSpace α inst✝⁸ : MeasurableSpace α' inst✝⁷ : MeasurableSpace β inst✝⁶ : MeasurableSpace β' inst✝⁵ : MeasurableSpace γ μ μ' : Measure α ν✝ ν' : Measure β τ : Measure γ inst✝⁴ : NormedAddCommGroup E inst✝³ : SigmaFinite ν✝ inst✝² : SigmaFinite μ ι : Type u_7 inst✝¹ : Finite ι ν : ι → Measure β inst✝ : ∀ (i : ι), SigmaFinite (ν i) ⊢ Measure.prod μ (sum ν) = sum fun i => Measure.prod μ (ν i) ** refine' prod_eq fun s t hs ht => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁹ : MeasurableSpace α inst✝⁸ : MeasurableSpace α' inst✝⁷ : MeasurableSpace β inst✝⁶ : MeasurableSpace β' inst✝⁵ : MeasurableSpace γ μ μ' : Measure α ν✝ ν' : Measure β τ : Measure γ inst✝⁴ : NormedAddCommGroup E inst✝³ : SigmaFinite ν✝ inst✝² : SigmaFinite μ ι : Type u_7 inst✝¹ : Finite ι ν : ι → Measure β inst✝ : ∀ (i : ι), SigmaFinite (ν i) s : Set α t : Set β hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑↑(sum fun i => Measure.prod μ (ν i)) (s ×ˢ t) = ↑↑μ s * ↑↑(sum ν) t ** simp_rw [sum_apply _ (hs.prod ht), sum_apply _ ht, prod_prod, ENNReal.tsum_mul_left] ** Qed
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MeasureTheory.Measure.sum_prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁹ : MeasurableSpace α inst✝⁸ : MeasurableSpace α' inst✝⁷ : MeasurableSpace β inst✝⁶ : MeasurableSpace β' inst✝⁵ : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁴ : NormedAddCommGroup E inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ✝ ι : Type u_7 inst✝¹ : Finite ι μ : ι → Measure α inst✝ : ∀ (i : ι), SigmaFinite (μ i) ⊢ Measure.prod (sum μ) ν = sum fun i => Measure.prod (μ i) ν ** refine' prod_eq fun s t hs ht => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁹ : MeasurableSpace α inst✝⁸ : MeasurableSpace α' inst✝⁷ : MeasurableSpace β inst✝⁶ : MeasurableSpace β' inst✝⁵ : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁴ : NormedAddCommGroup E inst✝³ : SigmaFinite ν inst✝² : SigmaFinite μ✝ ι : Type u_7 inst✝¹ : Finite ι μ : ι → Measure α inst✝ : ∀ (i : ι), SigmaFinite (μ i) s : Set α t : Set β hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑↑(sum fun i => Measure.prod (μ i) ν) (s ×ˢ t) = ↑↑(sum μ) s * ↑↑ν t ** simp_rw [sum_apply _ (hs.prod ht), sum_apply _ hs, prod_prod, ENNReal.tsum_mul_right] ** Qed
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MeasureTheory.Measure.prod_add ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ' : Measure α ν ν'✝ : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ ν' : Measure β inst✝ : SigmaFinite ν' ⊢ Measure.prod μ (ν + ν') = Measure.prod μ ν + Measure.prod μ ν' ** refine' prod_eq fun s t _ _ => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ' : Measure α ν ν'✝ : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ ν' : Measure β inst✝ : SigmaFinite ν' s : Set α t : Set β x✝¹ : MeasurableSet s x✝ : MeasurableSet t ⊢ ↑↑(Measure.prod μ ν + Measure.prod μ ν') (s ×ˢ t) = ↑↑μ s * ↑↑(ν + ν') t ** simp_rw [add_apply, prod_prod, left_distrib] ** Qed
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MeasureTheory.Measure.add_prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ'✝ : Measure α ν ν' : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ μ' : Measure α inst✝ : SigmaFinite μ' ⊢ Measure.prod (μ + μ') ν = Measure.prod μ ν + Measure.prod μ' ν ** refine' prod_eq fun s t _ _ => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ'✝ : Measure α ν ν' : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ μ' : Measure α inst✝ : SigmaFinite μ' s : Set α t : Set β x✝¹ : MeasurableSet s x✝ : MeasurableSet t ⊢ ↑↑(Measure.prod μ ν + Measure.prod μ' ν) (s ×ˢ t) = ↑↑(μ + μ') s * ↑↑ν t ** simp_rw [add_apply, prod_prod, right_distrib] ** Qed
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MeasureTheory.Measure.prod_zero ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ✝ μ : Measure α ⊢ Measure.prod μ 0 = 0 ** simp [Measure.prod] ** Qed
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MeasureTheory.Measure.map_prod_map ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ δ : Type u_7 inst✝ : MeasurableSpace δ f : α → β g : γ → δ μa : Measure α μc : Measure γ hfa : SigmaFinite (map f μa) hgc : SigmaFinite (map g μc) hf : Measurable f hg : Measurable g ⊢ Measure.prod (map f μa) (map g μc) = map (Prod.map f g) (Measure.prod μa μc) ** haveI := hgc.of_map μc hg.aemeasurable ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ δ : Type u_7 inst✝ : MeasurableSpace δ f : α → β g : γ → δ μa : Measure α μc : Measure γ hfa : SigmaFinite (map f μa) hgc : SigmaFinite (map g μc) hf : Measurable f hg : Measurable g this : SigmaFinite μc ⊢ Measure.prod (map f μa) (map g μc) = map (Prod.map f g) (Measure.prod μa μc) ** refine' prod_eq fun s t hs ht => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ δ : Type u_7 inst✝ : MeasurableSpace δ f : α → β g : γ → δ μa : Measure α μc : Measure γ hfa : SigmaFinite (map f μa) hgc : SigmaFinite (map g μc) hf : Measurable f hg : Measurable g this : SigmaFinite μc s : Set β t : Set δ hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑↑(map (Prod.map f g) (Measure.prod μa μc)) (s ×ˢ t) = ↑↑(map f μa) s * ↑↑(map g μc) t ** rw [map_apply (hf.prod_map hg) (hs.prod ht), map_apply hf hs, map_apply hg ht] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝³ : NormedAddCommGroup E inst✝² : SigmaFinite ν inst✝¹ : SigmaFinite μ δ : Type u_7 inst✝ : MeasurableSpace δ f : α → β g : γ → δ μa : Measure α μc : Measure γ hfa : SigmaFinite (map f μa) hgc : SigmaFinite (map g μc) hf : Measurable f hg : Measurable g this : SigmaFinite μc s : Set β t : Set δ hs : MeasurableSet s ht : MeasurableSet t ⊢ ↑↑(Measure.prod μa μc) (Prod.map f g ⁻¹' s ×ˢ t) = ↑↑μa (f ⁻¹' s) * ↑↑μc (g ⁻¹' t) ** exact prod_prod (f ⁻¹' s) (g ⁻¹' t) ** Qed
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MeasureTheory.QuasiMeasurePreserving.prod_of_right ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ✝ : Measure γ inst✝¹ : NormedAddCommGroup E f : α × β → γ μ : Measure α ν : Measure β τ : Measure γ hf : Measurable f inst✝ : SigmaFinite ν h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y) ⊢ QuasiMeasurePreserving f ** refine' ⟨hf, _⟩ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ✝ : Measure γ inst✝¹ : NormedAddCommGroup E f : α × β → γ μ : Measure α ν : Measure β τ : Measure γ hf : Measurable f inst✝ : SigmaFinite ν h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y) ⊢ map f (Measure.prod μ ν) ≪ τ ** refine' AbsolutelyContinuous.mk fun s hs h2s => _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ✝ : Measure γ inst✝¹ : NormedAddCommGroup E f : α × β → γ μ : Measure α ν : Measure β τ : Measure γ hf : Measurable f inst✝ : SigmaFinite ν h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y) s : Set γ hs : MeasurableSet s h2s : ↑↑τ s = 0 ⊢ ↑↑(map f (Measure.prod μ ν)) s = 0 ** rw [map_apply hf hs, prod_apply (hf hs)] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ✝ : Measure γ inst✝¹ : NormedAddCommGroup E f : α × β → γ μ : Measure α ν : Measure β τ : Measure γ hf : Measurable f inst✝ : SigmaFinite ν h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y) s : Set γ hs : MeasurableSet s h2s : ↑↑τ s = 0 ⊢ ∫⁻ (x : α), ↑↑ν (Prod.mk x ⁻¹' (f ⁻¹' s)) ∂μ = 0 ** simp_rw [preimage_preimage] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν✝ ν' : Measure β τ✝ : Measure γ inst✝¹ : NormedAddCommGroup E f : α × β → γ μ : Measure α ν : Measure β τ : Measure γ hf : Measurable f inst✝ : SigmaFinite ν h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y) s : Set γ hs : MeasurableSet s h2s : ↑↑τ s = 0 ⊢ ∫⁻ (x : α), ↑↑ν ((fun x_1 => f (x, x_1)) ⁻¹' s) ∂μ = 0 ** rw [lintegral_congr_ae (h2f.mono fun x hx => hx.preimage_null h2s), lintegral_zero] ** Qed
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AEMeasurable.prod_swap ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite μ inst✝ : SigmaFinite ν f : β × α → γ hf : AEMeasurable f ⊢ AEMeasurable fun z => f (Prod.swap z) ** exact hf.comp_measurable measurable_swap ** Qed
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MeasureTheory.lintegral_prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (z : α × β), f z ∂Measure.prod μ ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ ** have A : ∫⁻ z, f z ∂μ.prod ν = ∫⁻ z, hf.mk f z ∂μ.prod ν := lintegral_congr_ae hf.ae_eq_mk ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : α × β), f z ∂Measure.prod μ ν = ∫⁻ (z : α × β), AEMeasurable.mk f hf z ∂Measure.prod μ ν ⊢ ∫⁻ (z : α × β), f z ∂Measure.prod μ ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ ** have B : (∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ) = ∫⁻ x, ∫⁻ y, hf.mk f (x, y) ∂ν ∂μ := by
apply lintegral_congr_ae
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ ha using lintegral_congr_ae ha ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : α × β), f z ∂Measure.prod μ ν = ∫⁻ (z : α × β), AEMeasurable.mk f hf z ∂Measure.prod μ ν B : ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), AEMeasurable.mk f hf (x, y) ∂ν ∂μ ⊢ ∫⁻ (z : α × β), f z ∂Measure.prod μ ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ ** rw [A, B, lintegral_prod_of_measurable _ hf.measurable_mk] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : α × β), f z ∂Measure.prod μ ν = ∫⁻ (z : α × β), AEMeasurable.mk f hf z ∂Measure.prod μ ν ⊢ ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), AEMeasurable.mk f hf (x, y) ∂ν ∂μ ** apply lintegral_congr_ae ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : α × β), f z ∂Measure.prod μ ν = ∫⁻ (z : α × β), AEMeasurable.mk f hf z ∂Measure.prod μ ν ⊢ (fun a => ∫⁻ (y : β), f (a, y) ∂ν) =ᶠ[ae μ] fun a => ∫⁻ (y : β), AEMeasurable.mk f hf (a, y) ∂ν ** filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ ha using lintegral_congr_ae ha ** Qed
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MeasureTheory.lintegral_prod_mul ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α → ℝ≥0∞ g : β → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g ⊢ ∫⁻ (z : α × β), f z.1 * g z.2 ∂Measure.prod μ ν = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν ** simp [lintegral_prod _ (hf.fst.mul hg.snd), lintegral_lintegral_mul hf hg] ** Qed
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MeasureTheory.Measure.fst_apply ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) s : Set α hs : MeasurableSet s ⊢ ↑↑(fst ρ) s = ↑↑ρ (Prod.fst ⁻¹' s) ** rw [fst, Measure.map_apply measurable_fst hs] ** Qed
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MeasureTheory.Measure.fst_univ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) ⊢ ↑↑(fst ρ) univ = ↑↑ρ univ ** rw [fst_apply MeasurableSet.univ, preimage_univ] ** Qed
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MeasureTheory.Measure.fst_zero ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) ⊢ fst 0 = 0 ** simp [fst] ** Qed
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MeasureTheory.Measure.fst_map_prod_mk₀ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hY : AEMeasurable Y ⊢ fst (map (fun a => (X a, Y a)) μ) = map X μ ** by_cases hX : AEMeasurable X μ ** case pos α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hY : AEMeasurable Y hX : AEMeasurable X ⊢ fst (map (fun a => (X a, Y a)) μ) = map X μ ** ext1 s hs ** case pos.h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hY : AEMeasurable Y hX : AEMeasurable X s : Set β hs : MeasurableSet s ⊢ ↑↑(fst (map (fun a => (X a, Y a)) μ)) s = ↑↑(map X μ) s ** rw [Measure.fst_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_fst hs),
Measure.map_apply_of_aemeasurable hX hs, ← prod_univ, mk_preimage_prod, preimage_univ,
inter_univ] ** case neg α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hY : AEMeasurable Y hX : ¬AEMeasurable X ⊢ fst (map (fun a => (X a, Y a)) μ) = map X μ ** have : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ := by
contrapose! hX; exact measurable_fst.comp_aemeasurable hX ** case neg α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hY : AEMeasurable Y hX : ¬AEMeasurable X this : ¬AEMeasurable fun x => (X x, Y x) ⊢ fst (map (fun a => (X a, Y a)) μ) = map X μ ** simp [map_of_not_aemeasurable, hX, this] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hY : AEMeasurable Y hX : ¬AEMeasurable X ⊢ ¬AEMeasurable fun x => (X x, Y x) ** contrapose! hX ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hY : AEMeasurable Y hX : AEMeasurable fun x => (X x, Y x) ⊢ AEMeasurable X ** exact measurable_fst.comp_aemeasurable hX ** Qed
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MeasureTheory.Measure.snd_apply ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) s : Set β hs : MeasurableSet s ⊢ ↑↑(snd ρ) s = ↑↑ρ (Prod.snd ⁻¹' s) ** rw [snd, Measure.map_apply measurable_snd hs] ** Qed
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MeasureTheory.Measure.snd_univ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) ⊢ ↑↑(snd ρ) univ = ↑↑ρ univ ** rw [snd_apply MeasurableSet.univ, preimage_univ] ** Qed
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MeasureTheory.Measure.snd_zero ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) ⊢ snd 0 = 0 ** simp [snd] ** Qed
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MeasureTheory.Measure.snd_map_prod_mk₀ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hX : AEMeasurable X ⊢ snd (map (fun a => (X a, Y a)) μ) = map Y μ ** by_cases hY : AEMeasurable Y μ ** case pos α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hX : AEMeasurable X hY : AEMeasurable Y ⊢ snd (map (fun a => (X a, Y a)) μ) = map Y μ ** ext1 s hs ** case pos.h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hX : AEMeasurable X hY : AEMeasurable Y s : Set γ hs : MeasurableSet s ⊢ ↑↑(snd (map (fun a => (X a, Y a)) μ)) s = ↑↑(map Y μ) s ** rw [Measure.snd_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_snd hs),
Measure.map_apply_of_aemeasurable hY hs, ← univ_prod, mk_preimage_prod, preimage_univ,
univ_inter] ** case neg α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hX : AEMeasurable X hY : ¬AEMeasurable Y ⊢ snd (map (fun a => (X a, Y a)) μ) = map Y μ ** have : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ := by
contrapose! hY; exact measurable_snd.comp_aemeasurable hY ** case neg α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hX : AEMeasurable X hY : ¬AEMeasurable Y this : ¬AEMeasurable fun x => (X x, Y x) ⊢ snd (map (fun a => (X a, Y a)) μ) = map Y μ ** simp [map_of_not_aemeasurable, hY, this] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hX : AEMeasurable X hY : ¬AEMeasurable Y ⊢ ¬AEMeasurable fun x => (X x, Y x) ** contrapose! hY ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ✝ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν ρ : Measure (α × β) X : α → β Y : α → γ μ : Measure α hX : AEMeasurable X hY : AEMeasurable fun x => (X x, Y x) ⊢ AEMeasurable Y ** exact measurable_snd.comp_aemeasurable hY ** Qed
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aemeasurable_withDensity_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g : α → E ⊢ AEMeasurable g ↔ AEMeasurable fun x => ↑(f x) • g x ** constructor ** case mp α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g : α → E ⊢ AEMeasurable g → AEMeasurable fun x => ↑(f x) • g x ** rintro ⟨g', g'meas, hg'⟩ ** case mp.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' ⊢ AEMeasurable fun x => ↑(f x) • g x ** have A : MeasurableSet { x : α | f x ≠ 0 } := (hf (measurableSet_singleton 0)).compl ** case mp.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' A : MeasurableSet {x | f x ≠ 0} ⊢ AEMeasurable fun x => ↑(f x) • g x ** refine' ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.smul g'meas, _⟩ ** case mp.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' A : MeasurableSet {x | f x ≠ 0} ⊢ (fun x => ↑(f x) • g x) =ᵐ[μ] fun x => ↑(f x) • g' x ** apply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ { x | f x ≠ 0 } ** case mp.intro.intro.ht α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f x ≠ 0}, (fun x => ↑(f x) • g x) x = (fun x => ↑(f x) • g' x) x ** rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg' ** case mp.intro.intro.ht α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f x ≠ 0}, (fun x => ↑(f x) • g x) x = (fun x => ↑(f x) • g' x) x ** rw [ae_restrict_iff' A] ** case mp.intro.intro.ht α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | f x ≠ 0} → (fun x => ↑(f x) • g x) x = (fun x => ↑(f x) • g' x) x ** filter_upwards [hg'] ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ (a : α), (↑(f a) ≠ 0 → g a = g' a) → f a ≠ 0 → ↑(f a) • g a = ↑(f a) • g' a ** intro a ha h'a ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} a : α ha : ↑(f a) ≠ 0 → g a = g' a h'a : f a ≠ 0 ⊢ ↑(f a) • g a = ↑(f a) • g' a ** have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, coe_eq_zero] using h'a ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} a : α ha : ↑(f a) ≠ 0 → g a = g' a h'a : f a ≠ 0 this : ↑(f a) ≠ 0 ⊢ ↑(f a) • g a = ↑(f a) • g' a ** rw [ha this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} a : α ha : ↑(f a) ≠ 0 → g a = g' a h'a : f a ≠ 0 ⊢ ↑(f a) ≠ 0 ** simpa only [Ne.def, coe_eq_zero] using h'a ** case mp.intro.intro.htc α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f x ≠ 0}ᶜ, (fun x => ↑(f x) • g x) x = (fun x => ↑(f x) • g' x) x ** filter_upwards [ae_restrict_mem A.compl] ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ (a : α), a ∈ {x | f x ≠ 0}ᶜ → ↑(f a) • g a = ↑(f a) • g' a ** intro x hx ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' A : MeasurableSet {x | f x ≠ 0} x : α hx : x ∈ {x | f x ≠ 0}ᶜ ⊢ ↑(f x) • g x = ↑(f x) • g' x ** simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] g' A : MeasurableSet {x | f x ≠ 0} x : α hx : f x = 0 ⊢ ↑(f x) • g x = ↑(f x) • g' x ** simp [hx] ** case mpr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g : α → E ⊢ (AEMeasurable fun x => ↑(f x) • g x) → AEMeasurable g ** rintro ⟨g', g'meas, hg'⟩ ** case mpr.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : (fun x => ↑(f x) • g x) =ᵐ[μ] g' ⊢ AEMeasurable g ** refine' ⟨fun x => (f x : ℝ)⁻¹ • g' x, hf.coe_nnreal_real.inv.smul g'meas, _⟩ ** case mpr.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : (fun x => ↑(f x) • g x) =ᵐ[μ] g' ⊢ g =ᵐ[Measure.withDensity μ fun x => ↑(f x)] fun x => (↑(f x))⁻¹ • g' x ** rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] ** case mpr.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : (fun x => ↑(f x) • g x) =ᵐ[μ] g' ⊢ ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = (↑(f x))⁻¹ • g' x ** filter_upwards [hg'] ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : (fun x => ↑(f x) • g x) =ᵐ[μ] g' ⊢ ∀ (a : α), ↑(f a) • g a = g' a → ↑(f a) ≠ 0 → g a = (↑(f a))⁻¹ • g' a ** intro x hx h'x ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : (fun x => ↑(f x) • g x) =ᵐ[μ] g' x : α hx : ↑(f x) • g x = g' x h'x : ↑(f x) ≠ 0 ⊢ g x = (↑(f x))⁻¹ • g' x ** rw [← hx, smul_smul, _root_.inv_mul_cancel, one_smul] ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : (fun x => ↑(f x) • g x) =ᵐ[μ] g' x : α hx : ↑(f x) • g x = g' x h'x : ↑(f x) ≠ 0 ⊢ ↑(f x) ≠ 0 ** simp only [Ne.def, coe_eq_zero] at h'x ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ : Measure α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : TopologicalSpace.SecondCountableTopology E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E f : α → ℝ≥0 hf : Measurable f g g' : α → E g'meas : Measurable g' hg' : (fun x => ↑(f x) • g x) =ᵐ[μ] g' x : α hx : ↑(f x) • g x = g' x h'x : ¬f x = 0 ⊢ ↑(f x) ≠ 0 ** simpa only [NNReal.coe_eq_zero, Ne.def] using h'x ** Qed
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exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** rcases eq_empty_or_nonempty s with (rfl | hs) ** case inr E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** obtain ⟨T, T_count, hT⟩ :
∃ T : Set s,
T.Countable ∧ ⋃ x ∈ T, ball (f' (x : E)) (r (f' x)) = ⋃ x : s, ball (f' x) (r (f' x)) :=
TopologicalSpace.isOpen_iUnion_countable _ fun x => isOpen_ball ** case inr.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** obtain ⟨u, _, u_pos, u_lim⟩ :
∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
exists_seq_strictAnti_tendsto (0 : ℝ) ** case inr.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** let M : ℕ → T → Set E := fun n z =>
{x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - f' z (y - x)‖ ≤ r (f' z) * ‖y - x‖} ** case inr.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** have s_subset : ∀ x ∈ s, ∃ (n : ℕ) (z : T), x ∈ M n z := by
intro x xs
obtain ⟨z, zT, hz⟩ : ∃ z ∈ T, f' x ∈ ball (f' (z : E)) (r (f' z)) := by
have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by
rw [hT]
refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩
simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt
rwa [mem_iUnion₂, bex_def] at this
obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ ‖f' x - f' z‖ + ε ≤ r (f' z) := by
refine' ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩
simpa only [sub_pos] using mem_ball_iff_norm.mp hz
obtain ⟨δ, δpos, hδ⟩ :
∃ (δ : ℝ), 0 < δ ∧ ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=
Metric.mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) εpos)
obtain ⟨n, hn⟩ : ∃ n, u n < δ := ((tendsto_order.1 u_lim).2 _ δpos).exists
refine' ⟨n, ⟨z, zT⟩, ⟨xs, _⟩⟩
intro y hy
calc
‖f y - f x - (f' z) (y - x)‖ = ‖f y - f x - (f' x) (y - x) + (f' x - f' z) (y - x)‖ := by
congr 1
simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]
abel
_ ≤ ‖f y - f x - (f' x) (y - x)‖ + ‖(f' x - f' z) (y - x)‖ := (norm_add_le _ _)
_ ≤ ε * ‖y - x‖ + ‖f' x - f' z‖ * ‖y - x‖ := by
refine' add_le_add (hδ _) (ContinuousLinearMap.le_op_norm _ _)
rw [inter_comm]
exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy
_ ≤ r (f' z) * ‖y - x‖ := by
rw [← add_mul, add_comm]
exact mul_le_mul_of_nonneg_right hε (norm_nonneg _) ** case inr.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** have closure_M_subset : ∀ n z, s ∩ closure (M n z) ⊆ M n z := by
rintro n z x ⟨xs, hx⟩
refine' ⟨xs, fun y hy => _⟩
obtain ⟨a, aM, a_lim⟩ : ∃ a : ℕ → E, (∀ k, a k ∈ M n z) ∧ Tendsto a atTop (𝓝 x) :=
mem_closure_iff_seq_limit.1 hx
have L1 :
Tendsto (fun k : ℕ => ‖f y - f (a k) - (f' z) (y - a k)‖) atTop
(𝓝 ‖f y - f x - (f' z) (y - x)‖) := by
apply Tendsto.norm
have L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) := by
apply (hf' x xs).continuousWithinAt.tendsto.comp
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim
exact eventually_of_forall fun k => (aM k).1
apply Tendsto.sub (tendsto_const_nhds.sub L)
exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)
have L2 : Tendsto (fun k : ℕ => (r (f' z) : ℝ) * ‖y - a k‖) atTop (𝓝 (r (f' z) * ‖y - x‖)) :=
(tendsto_const_nhds.sub a_lim).norm.const_mul _
have I : ∀ᶠ k in atTop, ‖f y - f (a k) - (f' z) (y - a k)‖ ≤ r (f' z) * ‖y - a k‖ := by
have L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) :=
tendsto_const_nhds.dist a_lim
filter_upwards [(tendsto_order.1 L).2 _ hy.2]
intro k hk
exact (aM k).2 y ⟨hy.1, hk⟩
exact le_of_tendsto_of_tendsto L1 L2 I ** case inr.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** rcases TopologicalSpace.exists_dense_seq E with ⟨d, hd⟩ ** case inr.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** let K : ℕ → T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) ** case inr.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** have K_approx : ∀ (n) (z : T) (p), ApproximatesLinearOn f (f' z) (s ∩ K n z p) (r (f' z)) := by
intro n z p x hx y hy
have yM : y ∈ M n z := closure_M_subset _ _ ⟨hy.1, hy.2.1⟩
refine' yM.2 _ ⟨hx.1, _⟩
calc
dist x y ≤ dist x (d p) + dist y (d p) := dist_triangle_right _ _ _
_ ≤ u n / 3 + u n / 3 := (add_le_add hx.2.2 hy.2.2)
_ < u n := by linarith [u_pos n] ** case inr.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** have K_closed : ∀ (n) (z : T) (p), IsClosed (K n z p) := fun n z p =>
isClosed_closure.inter isClosed_ball ** case inr.intro.intro.intro.intro.intro.intro.intro E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (Set.Nonempty s → ∀ (n : ℕ), ∃ y, y ∈ s ∧ A n = f' y) ** refine'
⟨fun q => K (F q).1 (F q).2.1 (F q).2.2, fun q => f' (F q).2.1, fun n => K_closed _ _ _,
fun x xs => _, fun q => K_approx _ _ _, fun _ q => ⟨(F q).2.1, (F q).2.1.1.2, rfl⟩⟩ ** case inr.intro.intro.intro.intro.intro.intro.intro E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s ⊢ x ∈ ⋃ n, (fun q => K (F q).1 (F q).2.1 (F q).2.2) n ** obtain ⟨n, z, hnz⟩ : ∃ (n : ℕ) (z : T), x ∈ M n z := s_subset x xs ** case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z ⊢ x ∈ ⋃ n, (fun q => K (F q).1 (F q).2.1 (F q).2.2) n ** obtain ⟨p, hp⟩ : ∃ p : ℕ, x ∈ closedBall (d p) (u n / 3) := by
have : Set.Nonempty (ball x (u n / 3)) := by simp only [nonempty_ball]; linarith [u_pos n]
obtain ⟨p, hp⟩ : ∃ p : ℕ, d p ∈ ball x (u n / 3) := hd.exists_mem_open isOpen_ball this
exact ⟨p, (mem_ball'.1 hp).le⟩ ** case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z p : ℕ hp : x ∈ closedBall (d p) (u n / 3) ⊢ x ∈ ⋃ n, (fun q => K (F q).1 (F q).2.1 (F q).2.2) n ** obtain ⟨q, hq⟩ : ∃ q, F q = (n, z, p) := hF _ ** case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z p : ℕ hp : x ∈ closedBall (d p) (u n / 3) q : ℕ hq : F q = (n, z, p) ⊢ x ∈ ⋃ n, (fun q => K (F q).1 (F q).2.1 (F q).2.2) n ** apply mem_iUnion.2 ⟨q, _⟩ ** E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z p : ℕ hp : x ∈ closedBall (d p) (u n / 3) q : ℕ hq : F q = (n, z, p) ⊢ x ∈ (fun q => K (F q).1 (F q).2.1 (F q).2.2) q ** simp (config := { zeta := false }) only [hq, subset_closure hnz, hp, mem_inter_iff, and_true, hnz] ** case inl E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F f' : E → E →L[ℝ] F r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hf' : ∀ (x : E), x ∈ ∅ → HasFDerivWithinAt f (f' x) ∅ x ⊢ ∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ ∅ ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (∅ ∩ t n) (r (A n))) ∧ (Set.Nonempty ∅ → ∀ (n : ℕ), ∃ y, y ∈ ∅ ∧ A n = f' y) ** refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> simp ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} ⊢ ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z ** intro x xs ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s ⊢ ∃ n z, x ∈ M n z ** obtain ⟨z, zT, hz⟩ : ∃ z ∈ T, f' x ∈ ball (f' (z : E)) (r (f' z)) := by
have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by
rw [hT]
refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩
simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt
rwa [mem_iUnion₂, bex_def] at this ** case intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ⊢ ∃ n z, x ∈ M n z ** obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ ‖f' x - f' z‖ + ε ≤ r (f' z) := by
refine' ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩
simpa only [sub_pos] using mem_ball_iff_norm.mp hz ** case intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) ⊢ ∃ n z, x ∈ M n z ** obtain ⟨δ, δpos, hδ⟩ :
∃ (δ : ℝ), 0 < δ ∧ ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖} :=
Metric.mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) εpos) ** case intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} ⊢ ∃ n z, x ∈ M n z ** obtain ⟨n, hn⟩ : ∃ n, u n < δ := ((tendsto_order.1 u_lim).2 _ δpos).exists ** case intro.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ ⊢ ∃ n z, x ∈ M n z ** refine' ⟨n, ⟨z, zT⟩, ⟨xs, _⟩⟩ ** case intro.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ ⊢ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑{ val := z, property := zT }) (y - x)‖ ≤ ↑(r (f' ↑↑{ val := z, property := zT })) * ‖y - x‖ ** intro y hy ** case intro.intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ ‖f y - f x - ↑(f' ↑↑{ val := z, property := zT }) (y - x)‖ ≤ ↑(r (f' ↑↑{ val := z, property := zT })) * ‖y - x‖ ** calc
‖f y - f x - (f' z) (y - x)‖ = ‖f y - f x - (f' x) (y - x) + (f' x - f' z) (y - x)‖ := by
congr 1
simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]
abel
_ ≤ ‖f y - f x - (f' x) (y - x)‖ + ‖(f' x - f' z) (y - x)‖ := (norm_add_le _ _)
_ ≤ ε * ‖y - x‖ + ‖f' x - f' z‖ * ‖y - x‖ := by
refine' add_le_add (hδ _) (ContinuousLinearMap.le_op_norm _ _)
rw [inter_comm]
exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy
_ ≤ r (f' z) * ‖y - x‖ := by
rw [← add_mul, add_comm]
exact mul_le_mul_of_nonneg_right hε (norm_nonneg _) ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s ⊢ ∃ z, z ∈ T ∧ f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ** have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by
rw [hT]
refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩
simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s this : f' x ∈ ⋃ z ∈ T, ball (f' ↑z) ↑(r (f' ↑z)) ⊢ ∃ z, z ∈ T ∧ f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ** rwa [mem_iUnion₂, bex_def] at this ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s ⊢ f' x ∈ ⋃ z ∈ T, ball (f' ↑z) ↑(r (f' ↑z)) ** rw [hT] ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s ⊢ f' x ∈ ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) ** refine' mem_iUnion.2 ⟨⟨x, xs⟩, _⟩ ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s ⊢ f' x ∈ ball (f' ↑{ val := x, property := xs }) ↑(r (f' ↑{ val := x, property := xs })) ** simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ⊢ ∃ ε, 0 < ε ∧ ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) ** refine' ⟨r (f' z) - ‖f' x - f' z‖, _, le_of_eq (by abel)⟩ ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ⊢ 0 < ↑(r (f' ↑z)) - ‖f' x - f' ↑z‖ ** simpa only [sub_pos] using mem_ball_iff_norm.mp hz ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ⊢ ‖f' x - f' ↑z‖ + (↑(r (f' ↑z)) - ‖f' x - f' ↑z‖) = ↑(r (f' ↑z)) ** abel ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ ‖f y - f x - ↑(f' ↑z) (y - x)‖ = ‖f y - f x - ↑(f' x) (y - x) + ↑(f' x - f' ↑z) (y - x)‖ ** congr 1 ** case e_a E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ f y - f x - ↑(f' ↑z) (y - x) = f y - f x - ↑(f' x) (y - x) + ↑(f' x - f' ↑z) (y - x) ** simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply] ** case e_a E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ f y - f x - (↑(f' ↑z) y - ↑(f' ↑z) x) = f y - f x - (↑(f' x) y - ↑(f' x) x) + (↑(f' x) y - ↑(f' ↑z) y - (↑(f' x) x - ↑(f' ↑z) x)) ** abel ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ ‖f y - f x - ↑(f' x) (y - x)‖ + ‖↑(f' x - f' ↑z) (y - x)‖ ≤ ε * ‖y - x‖ + ‖f' x - f' ↑z‖ * ‖y - x‖ ** refine' add_le_add (hδ _) (ContinuousLinearMap.le_op_norm _ _) ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ y ∈ ball x δ ∩ s ** rw [inter_comm] ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ y ∈ s ∩ ball x δ ** exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ ε * ‖y - x‖ + ‖f' x - f' ↑z‖ * ‖y - x‖ ≤ ↑(r (f' ↑z)) * ‖y - x‖ ** rw [← add_mul, add_comm] ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} x : E xs : x ∈ s z : ↑s zT : z ∈ T hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z)) ε : ℝ εpos : 0 < ε hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z)) δ : ℝ δpos : 0 < δ hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - ↑(f' x) (y - x)‖ ≤ ε * ‖y - x‖} n : ℕ hn : u n < δ y : E hy : y ∈ s ∩ ball x (u n) ⊢ (‖f' x - f' ↑z‖ + ε) * ‖y - x‖ ≤ ↑(r (f' ↑z)) * ‖y - x‖ ** exact mul_le_mul_of_nonneg_right hε (norm_nonneg _) ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z ⊢ ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z ** rintro n z x ⟨xs, hx⟩ ** case intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) ⊢ x ∈ M n z ** refine' ⟨xs, fun y hy => _⟩ ** case intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) ⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖ ** obtain ⟨a, aM, a_lim⟩ : ∃ a : ℕ → E, (∀ k, a k ∈ M n z) ∧ Tendsto a atTop (𝓝 x) :=
mem_closure_iff_seq_limit.1 hx ** case intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) ⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖ ** have L1 :
Tendsto (fun k : ℕ => ‖f y - f (a k) - (f' z) (y - a k)‖) atTop
(𝓝 ‖f y - f x - (f' z) (y - x)‖) := by
apply Tendsto.norm
have L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) := by
apply (hf' x xs).continuousWithinAt.tendsto.comp
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim
exact eventually_of_forall fun k => (aM k).1
apply Tendsto.sub (tendsto_const_nhds.sub L)
exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim) ** case intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) ⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖ ** have L2 : Tendsto (fun k : ℕ => (r (f' z) : ℝ) * ‖y - a k‖) atTop (𝓝 (r (f' z) * ‖y - x‖)) :=
(tendsto_const_nhds.sub a_lim).norm.const_mul _ ** case intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) L2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖)) ⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖ ** have I : ∀ᶠ k in atTop, ‖f y - f (a k) - (f' z) (y - a k)‖ ≤ r (f' z) * ‖y - a k‖ := by
have L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) :=
tendsto_const_nhds.dist a_lim
filter_upwards [(tendsto_order.1 L).2 _ hy.2]
intro k hk
exact (aM k).2 y ⟨hy.1, hk⟩ ** case intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) L2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖)) I : ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖ ⊢ ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖ ** exact le_of_tendsto_of_tendsto L1 L2 I ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) ⊢ Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) ** apply Tendsto.norm ** case h E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) ⊢ Tendsto (fun x => f y - f (a x) - ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (f y - f x - ↑(f' ↑↑z) (y - x))) ** have L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) := by
apply (hf' x xs).continuousWithinAt.tendsto.comp
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim
exact eventually_of_forall fun k => (aM k).1 ** case h E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) ⊢ Tendsto (fun x => f y - f (a x) - ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (f y - f x - ↑(f' ↑↑z) (y - x))) ** apply Tendsto.sub (tendsto_const_nhds.sub L) ** case h E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L : Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) ⊢ Tendsto (fun x => ↑(f' ↑↑z) (y - a x)) atTop (𝓝 (↑(f' ↑↑z) (y - x))) ** exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim) ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) ⊢ Tendsto (fun k => f (a k)) atTop (𝓝 (f x)) ** apply (hf' x xs).continuousWithinAt.tendsto.comp ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) ⊢ Tendsto (fun k => a k) atTop (𝓝[s] x) ** apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) ⊢ ∀ᶠ (x : ℕ) in atTop, a x ∈ s ** exact eventually_of_forall fun k => (aM k).1 ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) L2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖)) ⊢ ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖ ** have L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) :=
tendsto_const_nhds.dist a_lim ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) L2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖)) L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) ⊢ ∀ᶠ (k : ℕ) in atTop, ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖ ** filter_upwards [(tendsto_order.1 L).2 _ hy.2] ** case h E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) L2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖)) L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) ⊢ ∀ (a_1 : ℕ), dist y (a a_1) < u n → ‖f y - f (a a_1) - ↑(f' ↑↑z) (y - a a_1)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a a_1‖ ** intro k hk ** case h E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z n : ℕ z : ↑T x : E xs : x ∈ s hx : x ∈ closure (M n z) y : E hy : y ∈ s ∩ ball x (u n) a : ℕ → E aM : ∀ (k : ℕ), a k ∈ M n z a_lim : Tendsto a atTop (𝓝 x) L1 : Tendsto (fun k => ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖) atTop (𝓝 ‖f y - f x - ↑(f' ↑↑z) (y - x)‖) L2 : Tendsto (fun k => ↑(r (f' ↑↑z)) * ‖y - a k‖) atTop (𝓝 (↑(r (f' ↑↑z)) * ‖y - x‖)) L : Tendsto (fun k => dist y (a k)) atTop (𝓝 (dist y x)) k : ℕ hk : dist y (a k) < u n ⊢ ‖f y - f (a k) - ↑(f' ↑↑z) (y - a k)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - a k‖ ** exact (aM k).2 y ⟨hy.1, hk⟩ ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) ⊢ ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) ** intro n z p x hx y hy ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) n : ℕ z : ↑T p : ℕ x : E hx : x ∈ s ∩ K n z p y : E hy : y ∈ s ∩ K n z p ⊢ ‖f x - f y - ↑(f' ↑↑z) (x - y)‖ ≤ ↑(r (f' ↑↑z)) * ‖x - y‖ ** have yM : y ∈ M n z := closure_M_subset _ _ ⟨hy.1, hy.2.1⟩ ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) n : ℕ z : ↑T p : ℕ x : E hx : x ∈ s ∩ K n z p y : E hy : y ∈ s ∩ K n z p yM : y ∈ M n z ⊢ ‖f x - f y - ↑(f' ↑↑z) (x - y)‖ ≤ ↑(r (f' ↑↑z)) * ‖x - y‖ ** refine' yM.2 _ ⟨hx.1, _⟩ ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) n : ℕ z : ↑T p : ℕ x : E hx : x ∈ s ∩ K n z p y : E hy : y ∈ s ∩ K n z p yM : y ∈ M n z ⊢ x ∈ ball y (u n) ** calc
dist x y ≤ dist x (d p) + dist y (d p) := dist_triangle_right _ _ _
_ ≤ u n / 3 + u n / 3 := (add_le_add hx.2.2 hy.2.2)
_ < u n := by linarith [u_pos n] ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) n : ℕ z : ↑T p : ℕ x : E hx : x ∈ s ∩ K n z p y : E hy : y ∈ s ∩ K n z p yM : y ∈ M n z ⊢ u n / 3 + u n / 3 < u n ** linarith [u_pos n] ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) ⊢ ∃ F, Function.Surjective F ** haveI : Encodable T := T_count.toEncodable ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) this✝ : Encodable ↑T this : Nonempty ↑T ⊢ ∃ F, Function.Surjective F ** inhabit ↥T ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) this✝ : Encodable ↑T this : Nonempty ↑T inhabited_h : Inhabited ↑T ⊢ ∃ F, Function.Surjective F ** exact ⟨_, Encodable.surjective_decode_iget (ℕ × T × ℕ)⟩ ** E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) this : Encodable ↑T ⊢ Nonempty ↑T ** rcases eq_empty_or_nonempty T with (rfl | hT) ** case inl E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) d : ℕ → E hd : DenseRange d T_count : Set.Countable ∅ hT : ⋃ x ∈ ∅, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) M : ℕ → ↑∅ → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z K : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p) this : Encodable ↑∅ ⊢ Nonempty ↑∅ ** rcases hs with ⟨x, xs⟩ ** case inl.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) d : ℕ → E hd : DenseRange d T_count : Set.Countable ∅ hT : ⋃ x ∈ ∅, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) M : ℕ → ↑∅ → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z K : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p) this : Encodable ↑∅ x : E xs : x ∈ s ⊢ Nonempty ↑∅ ** rcases s_subset x xs with ⟨n, z, _⟩ ** case inl.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) d : ℕ → E hd : DenseRange d T_count : Set.Countable ∅ hT : ⋃ x ∈ ∅, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) M : ℕ → ↑∅ → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑∅), s ∩ closure (M n z) ⊆ M n z K : ℕ → ↑∅ → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑∅) (p : ℕ), IsClosed (K n z p) this : Encodable ↑∅ x : E xs : x ∈ s n : ℕ z : ↑∅ h✝ : x ∈ M n z ⊢ Nonempty ↑∅ ** exact False.elim z.2 ** case inr E : Type u_1 F : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F f : E → F s : Set E f' : E → E →L[ℝ] F hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT✝ : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) this : Encodable ↑T hT : Set.Nonempty T ⊢ Nonempty ↑T ** exact hT.coe_sort ** E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z ⊢ ∃ p, x ∈ closedBall (d p) (u n / 3) ** have : Set.Nonempty (ball x (u n / 3)) := by simp only [nonempty_ball]; linarith [u_pos n] ** E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z this : Set.Nonempty (ball x (u n / 3)) ⊢ ∃ p, x ∈ closedBall (d p) (u n / 3) ** obtain ⟨p, hp⟩ : ∃ p : ℕ, d p ∈ ball x (u n / 3) := hd.exists_mem_open isOpen_ball this ** case intro E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z this : Set.Nonempty (ball x (u n / 3)) p : ℕ hp : d p ∈ ball x (u n / 3) ⊢ ∃ p, x ∈ closedBall (d p) (u n / 3) ** exact ⟨p, (mem_ball'.1 hp).le⟩ ** E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z ⊢ Set.Nonempty (ball x (u n / 3)) ** simp only [nonempty_ball] ** E : Type u_1 F✝ : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : FiniteDimensional ℝ E inst✝² : NormedAddCommGroup F✝ inst✝¹ : NormedSpace ℝ F✝ s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝ : SecondCountableTopology F✝ f : E → F✝ s : Set E f' : E → E →L[ℝ] F✝ hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x r : (E →L[ℝ] F✝) → ℝ≥0 rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0 hs : Set.Nonempty s T : Set ↑s T_count : Set.Countable T hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x)) u : ℕ → ℝ left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ (y : E), y ∈ s ∩ ball x (u n) → ‖f y - f x - ↑(f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖} s_subset : ∀ (x : E), x ∈ s → ∃ n z, x ∈ M n z closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z d : ℕ → E hd : DenseRange d K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3) K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z)) K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p) F : ℕ → ℕ × ↑T × ℕ hF : Function.Surjective F x : E xs : x ∈ s n : ℕ z : ↑T hnz : x ∈ M n z ⊢ 0 < u n / 3 ** linarith [u_pos n] ** Qed
| |
MeasureTheory.addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 ⊢ ↑↑μ (f '' s) = 0 ** refine' le_antisymm _ (zero_le _) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 ⊢ ↑↑μ (f '' s) ≤ 0 ** have :
∀ A : E →L[ℝ] E, ∃ δ : ℝ≥0, 0 < δ ∧
∀ (t : Set E), ApproximatesLinearOn f A t δ →
μ (f '' t) ≤ (Real.toNNReal |A.det| + 1 : ℝ≥0) * μ t := by
intro A
let m : ℝ≥0 := Real.toNNReal |A.det| + 1
have I : ENNReal.ofReal |A.det| < m := by
simp only [ENNReal.ofReal, lt_add_iff_pos_right, zero_lt_one, ENNReal.coe_lt_coe]
rcases ((addHaar_image_le_mul_of_det_lt μ A I).and self_mem_nhdsWithin).exists with ⟨δ, h, h'⟩
exact ⟨δ, h', fun t ht => h t f ht⟩ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 this : ∀ (A : E →L[ℝ] E), ∃ δ, 0 < δ ∧ ∀ (t : Set E), ApproximatesLinearOn f A t δ → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t ⊢ ↑↑μ (f '' s) ≤ 0 ** choose δ hδ using this ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t ⊢ ↑↑μ (f '' s) ≤ 0 ** obtain ⟨t, A, _, _, t_cover, ht, -⟩ :
∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] E),
Pairwise (Disjoint on t) ∧
(∀ n : ℕ, MeasurableSet (t n)) ∧
(s ⊆ ⋃ n : ℕ, t n) ∧
(∀ n : ℕ, ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))) ∧
(s.Nonempty → ∀ n, ∃ y ∈ s, A n = fderivWithin ℝ f s y) :=
exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s (fderivWithin ℝ f s)
(fun x xs => (hf x xs).hasFDerivWithinAt) δ fun A => (hδ A).1.ne' ** case intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ↑↑μ (f '' s) ≤ 0 ** calc
μ (f '' s) ≤ μ (⋃ n, f '' (s ∩ t n)) := by
apply measure_mono
rw [← image_iUnion, ← inter_iUnion]
exact image_subset f (subset_inter Subset.rfl t_cover)
_ ≤ ∑' n, μ (f '' (s ∩ t n)) := (measure_iUnion_le _)
_ ≤ ∑' n, (Real.toNNReal |(A n).det| + 1 : ℝ≥0) * μ (s ∩ t n) := by
apply ENNReal.tsum_le_tsum fun n => ?_
apply (hδ (A n)).2
exact ht n
_ ≤ ∑' n, ((Real.toNNReal |(A n).det| + 1 : ℝ≥0) : ℝ≥0∞) * 0 := by
refine' ENNReal.tsum_le_tsum fun n => mul_le_mul_left' _ _
exact le_trans (measure_mono (inter_subset_left _ _)) (le_of_eq hs)
_ = 0 := by simp only [tsum_zero, mul_zero] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 ⊢ ∀ (A : E →L[ℝ] E), ∃ δ, 0 < δ ∧ ∀ (t : Set E), ApproximatesLinearOn f A t δ → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t ** intro A ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 A : E →L[ℝ] E ⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E), ApproximatesLinearOn f A t δ → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t ** let m : ℝ≥0 := Real.toNNReal |A.det| + 1 ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + 1 ⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E), ApproximatesLinearOn f A t δ → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t ** have I : ENNReal.ofReal |A.det| < m := by
simp only [ENNReal.ofReal, lt_add_iff_pos_right, zero_lt_one, ENNReal.coe_lt_coe] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + 1 I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m ⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E), ApproximatesLinearOn f A t δ → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t ** rcases ((addHaar_image_le_mul_of_det_lt μ A I).and self_mem_nhdsWithin).exists with ⟨δ, h, h'⟩ ** case intro.intro E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + 1 I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s h' : 0 < δ ⊢ ∃ δ, 0 < δ ∧ ∀ (t : Set E), ApproximatesLinearOn f A t δ → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t ** exact ⟨δ, h', fun t ht => h t f ht⟩ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + 1 ⊢ ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m ** simp only [ENNReal.ofReal, lt_add_iff_pos_right, zero_lt_one, ENNReal.coe_lt_coe] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ↑↑μ (f '' s) ≤ ↑↑μ (⋃ n, f '' (s ∩ t n)) ** apply measure_mono ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ f '' s ⊆ ⋃ n, f '' (s ∩ t n) ** rw [← image_iUnion, ← inter_iUnion] ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ f '' s ⊆ f '' (s ∩ ⋃ i, t i) ** exact image_subset f (subset_inter Subset.rfl t_cover) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∑' (n : ℕ), ↑↑μ (f '' (s ∩ t n)) ≤ ∑' (n : ℕ), ↑(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * ↑↑μ (s ∩ t n) ** apply ENNReal.tsum_le_tsum fun n => ?_ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ↑↑μ (f '' (s ∩ t n)) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * ↑↑μ (s ∩ t n) ** apply (hδ (A n)).2 ** case a E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ** exact ht n ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∑' (n : ℕ), ↑(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * ↑↑μ (s ∩ t n) ≤ ∑' (n : ℕ), ↑(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * 0 ** refine' ENNReal.tsum_le_tsum fun n => mul_le_mul_left' _ _ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ↑↑μ (s ∩ t n) ≤ 0 ** exact le_trans (measure_mono (inter_subset_left _ _)) (le_of_eq hs) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hf : DifferentiableOn ℝ f s hs : ↑↑μ s = 0 δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ ∀ (t : Set E), ApproximatesLinearOn f A t (δ A) → ↑↑μ (f '' t) ≤ ↑(Real.toNNReal |ContinuousLinearMap.det A| + 1) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E left✝¹ : Pairwise (Disjoint on t) left✝ : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∑' (n : ℕ), ↑(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * 0 = 0 ** simp only [tsum_zero, mul_zero] ** Qed
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MeasureTheory.aemeasurable_ofReal_abs_det_fderivWithin ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)| ** apply ENNReal.measurable_ofReal.comp_aemeasurable ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => |ContinuousLinearMap.det (f' x)| ** refine' continuous_abs.measurable.comp_aemeasurable _ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => ContinuousLinearMap.det (f' x) ** refine' ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable _ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => f' x ** exact aemeasurable_fderivWithin μ hs hf' ** Qed
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MeasureTheory.aemeasurable_toNNReal_abs_det_fderivWithin ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => Real.toNNReal |ContinuousLinearMap.det (f' x)| ** apply measurable_real_toNNReal.comp_aemeasurable ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => |ContinuousLinearMap.det (f' x)| ** refine' continuous_abs.measurable.comp_aemeasurable _ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => ContinuousLinearMap.det (f' x) ** refine' ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable _ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ⊢ AEMeasurable fun x => f' x ** exact aemeasurable_fderivWithin μ hs hf' ** Qed
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MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux1 ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε this : ∀ (A : E →L[ℝ] E), ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| ∂μ + 2 * ↑ε * ↑↑μ s ** choose δ hδ using this ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| ∂μ + 2 * ↑ε * ↑↑μ s ** obtain ⟨t, A, t_disj, t_meas, t_cover, ht, -⟩ :
∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] E),
Pairwise (Disjoint on t) ∧
(∀ n : ℕ, MeasurableSet (t n)) ∧
(s ⊆ ⋃ n : ℕ, t n) ∧
(∀ n : ℕ, ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))) ∧
(s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) :=
exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' δ fun A => (hδ A).1.ne' ** case intro.intro.intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| ∂μ + 2 * ↑ε * ↑↑μ s ** calc
μ (f '' s) ≤ μ (⋃ n, f '' (s ∩ t n)) := by
apply measure_mono
rw [← image_iUnion, ← inter_iUnion]
exact image_subset f (subset_inter Subset.rfl t_cover)
_ ≤ ∑' n, μ (f '' (s ∩ t n)) := (measure_iUnion_le _)
_ ≤ ∑' n, (ENNReal.ofReal |(A n).det| + ε) * μ (s ∩ t n) := by
apply ENNReal.tsum_le_tsum fun n => ?_
apply (hδ (A n)).2.2
exact ht n
_ = ∑' n, ∫⁻ _ in s ∩ t n, ENNReal.ofReal |(A n).det| + ε ∂μ := by
simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
_ ≤ ∑' n, ∫⁻ x in s ∩ t n, ENNReal.ofReal |(f' x).det| + 2 * ε ∂μ := by
apply ENNReal.tsum_le_tsum fun n => ?_
apply lintegral_mono_ae
filter_upwards [(ht n).norm_fderiv_sub_le μ (hs.inter (t_meas n)) f' fun x hx =>
(hf' x hx.1).mono (inter_subset_left _ _)]
intro x hx
have I : |(A n).det| ≤ |(f' x).det| + ε :=
calc
|(A n).det| = |(f' x).det - ((f' x).det - (A n).det)| := by congr 1; abel
_ ≤ |(f' x).det| + |(f' x).det - (A n).det| := (abs_sub _ _)
_ ≤ |(f' x).det| + ε := add_le_add le_rfl ((hδ (A n)).2.1 _ hx)
calc
ENNReal.ofReal |(A n).det| + ε ≤ ENNReal.ofReal (|(f' x).det| + ε) + ε :=
add_le_add (ENNReal.ofReal_le_ofReal I) le_rfl
_ = ENNReal.ofReal |(f' x).det| + 2 * ε := by
simp only [ENNReal.ofReal_add, abs_nonneg, two_mul, add_assoc, NNReal.zero_le_coe,
ENNReal.ofReal_coe_nnreal]
_ = ∫⁻ x in ⋃ n, s ∩ t n, ENNReal.ofReal |(f' x).det| + 2 * ε ∂μ := by
have M : ∀ n : ℕ, MeasurableSet (s ∩ t n) := fun n => hs.inter (t_meas n)
rw [lintegral_iUnion M]
exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _
_ = ∫⁻ x in s, ENNReal.ofReal |(f' x).det| + 2 * ε ∂μ := by
have : s = ⋃ n, s ∩ t n := by
rw [← inter_iUnion]
exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)
rw [← this]
_ = (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) + 2 * ε * μ s := by
simp only [lintegral_add_right' _ aemeasurable_const, set_lintegral_const] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε ⊢ ∀ (A : E →L[ℝ] E), ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** intro A ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E ⊢ ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** let m : ℝ≥0 := Real.toNNReal |A.det| + ε ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε ⊢ ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** have I : ENNReal.ofReal |A.det| < m := by
simp only [ENNReal.ofReal, lt_add_iff_pos_right, εpos, ENNReal.coe_lt_coe] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m ⊢ ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** rcases ((addHaar_image_le_mul_of_det_lt μ A I).and self_mem_nhdsWithin).exists with ⟨δ, h, δpos⟩ ** case intro.intro E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ ⊢ ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** obtain ⟨δ', δ'pos, hδ'⟩ : ∃ (δ' : ℝ), 0 < δ' ∧ ∀ B, dist B A < δ' → dist B.det A.det < ↑ε :=
continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt ε εpos ** case intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε ⊢ ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** let δ'' : ℝ≥0 := ⟨δ' / 2, (half_pos δ'pos).le⟩ ** case intro.intro.intro.intro E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } ⊢ ∃ δ, 0 < δ ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑δ → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t δ → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** refine' ⟨min δ δ'', lt_min δpos (half_pos δ'pos), _, _⟩ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε ⊢ ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m ** simp only [ENNReal.ofReal, lt_add_iff_pos_right, εpos, ENNReal.coe_lt_coe] ** case intro.intro.intro.intro.refine'_1 E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } ⊢ ∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(min δ δ'') → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε ** intro B hB ** case intro.intro.intro.intro.refine'_1 E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } B : E →L[ℝ] E hB : ‖B - A‖ ≤ ↑(min δ δ'') ⊢ |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε ** rw [← Real.dist_eq] ** case intro.intro.intro.intro.refine'_1 E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } B : E →L[ℝ] E hB : ‖B - A‖ ≤ ↑(min δ δ'') ⊢ dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) ≤ ↑ε ** apply (hδ' B _).le ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } B : E →L[ℝ] E hB : ‖B - A‖ ≤ ↑(min δ δ'') ⊢ dist B A < δ' ** rw [dist_eq_norm] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } B : E →L[ℝ] E hB : ‖B - A‖ ≤ ↑(min δ δ'') ⊢ ‖B - A‖ < δ' ** calc
‖B - A‖ ≤ (min δ δ'' : ℝ≥0) := hB
_ ≤ δ'' := by simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]
_ < δ' := half_lt_self δ'pos ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } B : E →L[ℝ] E hB : ‖B - A‖ ≤ ↑(min δ δ'') ⊢ ↑(min δ δ'') ≤ ↑δ'' ** simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff] ** case intro.intro.intro.intro.refine'_2 E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } ⊢ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (min δ δ'') → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** intro t g htg ** case intro.intro.intro.intro.refine'_2 E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε A : E →L[ℝ] E m : ℝ≥0 := Real.toNNReal |ContinuousLinearMap.det A| + ε I : ENNReal.ofReal |ContinuousLinearMap.det A| < ↑m δ : ℝ≥0 h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑↑μ (f '' s) ≤ ↑m * ↑↑μ s δpos : 0 < δ δ' : ℝ δ'pos : 0 < δ' hδ' : ∀ (B : E →L[ℝ] E), dist B A < δ' → dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < ↑ε δ'' : ℝ≥0 := { val := δ' / 2, property := (_ : 0 ≤ δ' / 2) } t : Set E g : E → E htg : ApproximatesLinearOn g A t (min δ δ'') ⊢ ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t ** exact h t g (htg.mono_num (min_le_left _ _)) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ↑↑μ (f '' s) ≤ ↑↑μ (⋃ n, f '' (s ∩ t n)) ** apply measure_mono ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ f '' s ⊆ ⋃ n, f '' (s ∩ t n) ** rw [← image_iUnion, ← inter_iUnion] ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ f '' s ⊆ f '' (s ∩ ⋃ i, t i) ** exact image_subset f (subset_inter Subset.rfl t_cover) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∑' (n : ℕ), ↑↑μ (f '' (s ∩ t n)) ≤ ∑' (n : ℕ), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε) * ↑↑μ (s ∩ t n) ** apply ENNReal.tsum_le_tsum fun n => ?_ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ↑↑μ (f '' (s ∩ t n)) ≤ (ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε) * ↑↑μ (s ∩ t n) ** apply (hδ (A n)).2.2 ** case a E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ** exact ht n ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∑' (n : ℕ), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε) * ↑↑μ (s ∩ t n) = ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ∂μ ** simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ∂μ ≤ ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ ** apply ENNReal.tsum_le_tsum fun n => ?_ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ∂μ ≤ ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ ** apply lintegral_mono_ae ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ∀ᵐ (a : E) ∂Measure.restrict μ (s ∩ t n), ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ≤ ENNReal.ofReal |ContinuousLinearMap.det (f' a)| + 2 * ↑ε ** filter_upwards [(ht n).norm_fderiv_sub_le μ (hs.inter (t_meas n)) f' fun x hx =>
(hf' x hx.1).mono (inter_subset_left _ _)] ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ ⊢ ∀ (a : E), ‖f' a - A n‖₊ ≤ δ (A n) → ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ≤ ENNReal.ofReal |ContinuousLinearMap.det (f' a)| + 2 * ↑ε ** intro x hx ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ x : E hx : ‖f' x - A n‖₊ ≤ δ (A n) ⊢ ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ≤ ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ** have I : |(A n).det| ≤ |(f' x).det| + ε :=
calc
|(A n).det| = |(f' x).det - ((f' x).det - (A n).det)| := by congr 1; abel
_ ≤ |(f' x).det| + |(f' x).det - (A n).det| := (abs_sub _ _)
_ ≤ |(f' x).det| + ε := add_le_add le_rfl ((hδ (A n)).2.1 _ hx) ** case h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ x : E hx : ‖f' x - A n‖₊ ≤ δ (A n) I : |ContinuousLinearMap.det (A n)| ≤ |ContinuousLinearMap.det (f' x)| + ↑ε ⊢ ENNReal.ofReal |ContinuousLinearMap.det (A n)| + ↑ε ≤ ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ** calc
ENNReal.ofReal |(A n).det| + ε ≤ ENNReal.ofReal (|(f' x).det| + ε) + ε :=
add_le_add (ENNReal.ofReal_le_ofReal I) le_rfl
_ = ENNReal.ofReal |(f' x).det| + 2 * ε := by
simp only [ENNReal.ofReal_add, abs_nonneg, two_mul, add_assoc, NNReal.zero_le_coe,
ENNReal.ofReal_coe_nnreal] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ x : E hx : ‖f' x - A n‖₊ ≤ δ (A n) ⊢ |ContinuousLinearMap.det (A n)| = |ContinuousLinearMap.det (f' x) - (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))| ** congr 1 ** case e_a E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ x : E hx : ‖f' x - A n‖₊ ≤ δ (A n) ⊢ ContinuousLinearMap.det (A n) = ContinuousLinearMap.det (f' x) - (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n)) ** abel ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) n : ℕ x : E hx : ‖f' x - A n‖₊ ≤ δ (A n) I : |ContinuousLinearMap.det (A n)| ≤ |ContinuousLinearMap.det (f' x)| + ↑ε ⊢ ENNReal.ofReal (|ContinuousLinearMap.det (f' x)| + ↑ε) + ↑ε = ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ** simp only [ENNReal.ofReal_add, abs_nonneg, two_mul, add_assoc, NNReal.zero_le_coe,
ENNReal.ofReal_coe_nnreal] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ = ∫⁻ (x : E) in ⋃ n, s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ ** have M : ∀ n : ℕ, MeasurableSet (s ∩ t n) := fun n => hs.inter (t_meas n) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) M : ∀ (n : ℕ), MeasurableSet (s ∩ t n) ⊢ ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ = ∫⁻ (x : E) in ⋃ n, s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ ** rw [lintegral_iUnion M] ** case hd E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) M : ∀ (n : ℕ), MeasurableSet (s ∩ t n) ⊢ Pairwise (Disjoint on fun i => s ∩ t i) ** exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∫⁻ (x : E) in ⋃ n, s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ = ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ ** have : s = ⋃ n, s ∩ t n := by
rw [← inter_iUnion]
exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) this : s = ⋃ n, s ∩ t n ⊢ ∫⁻ (x : E) in ⋃ n, s ∩ t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ = ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ ** rw [← this] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ s = ⋃ n, s ∩ t n ** rw [← inter_iUnion] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ s = s ∩ ⋃ i, t i ** exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _) ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x ε : ℝ≥0 εpos : 0 < ε δ : (E →L[ℝ] E) → ℝ≥0 hδ : ∀ (A : E →L[ℝ] E), 0 < δ A ∧ (∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |ContinuousLinearMap.det B - ContinuousLinearMap.det A| ≤ ↑ε) ∧ ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g A t (δ A) → ↑↑μ (g '' t) ≤ (ENNReal.ofReal |ContinuousLinearMap.det A| + ↑ε) * ↑↑μ t t : ℕ → Set E A : ℕ → E →L[ℝ] E t_disj : Pairwise (Disjoint on t) t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_cover : s ⊆ ⋃ n, t n ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n)) ⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * ↑ε ∂μ = ∫⁻ (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| ∂μ + 2 * ↑ε * ↑↑μ s ** simp only [lintegral_add_right' _ aemeasurable_const, set_lintegral_const] ** Qed
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MeasureTheory.integrableOn_image_iff_integrableOn_abs_det_fderiv_smul ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F ⊢ IntegrableOn g (f '' s) ↔ IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| • g (f x)) s ** rw [IntegrableOn, ← restrict_map_withDensity_abs_det_fderiv_eq_addHaar μ hs hf' hf,
(measurableEmbedding_of_fderivWithin hs hf' hf).integrable_map_iff] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F ⊢ Integrable (g ∘ Set.restrict s f) ↔ IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| • g (f x)) s ** change Integrable ((g ∘ f) ∘ ((↑) : s → E)) _ ↔ _ ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F ⊢ Integrable ((g ∘ f) ∘ Subtype.val) ↔ IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| • g (f x)) s ** rw [← (MeasurableEmbedding.subtype_coe hs).integrable_map_iff, map_comap_subtype_coe hs] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F ⊢ Integrable (g ∘ f) ↔ IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| • g (f x)) s ** rw [restrict_withDensity hs, integrable_withDensity_iff_integrable_coe_smul₀, IntegrableOn] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F ⊢ (Integrable fun x => ↑(Real.toNNReal |ContinuousLinearMap.det (f' x)|) • (g ∘ f) x) ↔ Integrable fun x => |ContinuousLinearMap.det (f' x)| • g (f x) ** rw [iff_iff_eq] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F ⊢ (Integrable fun x => ↑(Real.toNNReal |ContinuousLinearMap.det (f' x)|) • (g ∘ f) x) = Integrable fun x => |ContinuousLinearMap.det (f' x)| • g (f x) ** congr 2 with x ** case e_f.h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F x : E ⊢ ↑(Real.toNNReal |ContinuousLinearMap.det (f' x)|) • (g ∘ f) x = |ContinuousLinearMap.det (f' x)| • g (f x) ** rw [Real.coe_toNNReal _ (abs_nonneg _)] ** case e_f.h E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F x : E ⊢ |ContinuousLinearMap.det (f' x)| • (g ∘ f) x = |ContinuousLinearMap.det (f' x)| • g (f x) ** rfl ** case hf E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ hs : MeasurableSet s hf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x hf : InjOn f s g : E → F ⊢ AEMeasurable fun x => Real.toNNReal |ContinuousLinearMap.det (f' x)| ** exact aemeasurable_toNNReal_abs_det_fderivWithin μ hs hf' ** Qed
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MeasureTheory.det_one_smulRight ** E : Type u_1 F : Type u_2 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : FiniteDimensional ℝ E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E μ : Measure E inst✝¹ : IsAddHaarMeasure μ 𝕜 : Type u_3 inst✝ : NormedField 𝕜 v : 𝕜 ⊢ ContinuousLinearMap.det (ContinuousLinearMap.smulRight 1 v) = v ** have : (1 : 𝕜 →L[𝕜] 𝕜).smulRight v = v • (1 : 𝕜 →L[𝕜] 𝕜) := by
ext1
simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one] ** E : Type u_1 F : Type u_2 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : FiniteDimensional ℝ E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E μ : Measure E inst✝¹ : IsAddHaarMeasure μ 𝕜 : Type u_3 inst✝ : NormedField 𝕜 v : 𝕜 this : ContinuousLinearMap.smulRight 1 v = v • 1 ⊢ ContinuousLinearMap.det (ContinuousLinearMap.smulRight 1 v) = v ** rw [this, ContinuousLinearMap.det, ContinuousLinearMap.coe_smul] ** E : Type u_1 F : Type u_2 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : FiniteDimensional ℝ E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E μ : Measure E inst✝¹ : IsAddHaarMeasure μ 𝕜 : Type u_3 inst✝ : NormedField 𝕜 v : 𝕜 this : ContinuousLinearMap.smulRight 1 v = v • 1 ⊢ ↑LinearMap.det (v • ↑1) = v ** rw [show ((1 : 𝕜 →L[𝕜] 𝕜) : 𝕜 →ₗ[𝕜] 𝕜) = LinearMap.id from rfl] ** E : Type u_1 F : Type u_2 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : FiniteDimensional ℝ E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E μ : Measure E inst✝¹ : IsAddHaarMeasure μ 𝕜 : Type u_3 inst✝ : NormedField 𝕜 v : 𝕜 this : ContinuousLinearMap.smulRight 1 v = v • 1 ⊢ ↑LinearMap.det (v • LinearMap.id) = v ** rw [LinearMap.det_smul, FiniteDimensional.finrank_self, LinearMap.det_id, pow_one, mul_one] ** E : Type u_1 F : Type u_2 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : FiniteDimensional ℝ E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E μ : Measure E inst✝¹ : IsAddHaarMeasure μ 𝕜 : Type u_3 inst✝ : NormedField 𝕜 v : 𝕜 ⊢ ContinuousLinearMap.smulRight 1 v = v • 1 ** ext1 ** case h E : Type u_1 F : Type u_2 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : FiniteDimensional ℝ E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F s : Set E f : E → E f' : E → E →L[ℝ] E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E μ : Measure E inst✝¹ : IsAddHaarMeasure μ 𝕜 : Type u_3 inst✝ : NormedField 𝕜 v : 𝕜 ⊢ ↑(ContinuousLinearMap.smulRight 1 v) 1 = ↑(v • 1) 1 ** simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one] ** Qed
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MeasureTheory.integral_image_eq_integral_abs_deriv_smul ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s✝ : Set E f✝ : E → E f'✝ : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ s : Set ℝ f f' : ℝ → ℝ hs : MeasurableSet s hf' : ∀ (x : ℝ), x ∈ s → HasDerivWithinAt f (f' x) s x hf : InjOn f s g : ℝ → F ⊢ ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, |f' x| • g (f x) ** simpa only [det_one_smulRight] using
integral_image_eq_integral_abs_det_fderiv_smul volume hs
(fun x hx => (hf' x hx).hasFDerivWithinAt) hf g ** Qed
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MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f✝ : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ f : LocalHomeomorph E E hf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x g : E → F ⊢ ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |ContinuousLinearMap.det (f' x)| • g (↑f x) ∂μ ** have : f '' f.source = f.target := LocalEquiv.image_source_eq_target f.toLocalEquiv ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f✝ : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ f : LocalHomeomorph E E hf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x g : E → F this : ↑f '' f.source = f.target ⊢ ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |ContinuousLinearMap.det (f' x)| • g (↑f x) ∂μ ** rw [← this] ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f✝ : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ f : LocalHomeomorph E E hf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x g : E → F this : ↑f '' f.source = f.target ⊢ ∫ (x : E) in ↑f '' f.source, g x ∂μ = ∫ (x : E) in f.source, |ContinuousLinearMap.det (f' x)| • g (↑f x) ∂μ ** apply integral_image_eq_integral_abs_det_fderiv_smul μ f.open_source.measurableSet _ f.injOn ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f✝ : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ f : LocalHomeomorph E E hf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x g : E → F this : ↑f '' f.source = f.target ⊢ ∀ (x : E), x ∈ f.source → HasFDerivWithinAt (↑f) (f' x) f.source x ** intro x hx ** E : Type u_1 F : Type u_2 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : FiniteDimensional ℝ E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F s : Set E f✝ : E → E f' : E → E →L[ℝ] E inst✝² : MeasurableSpace E inst✝¹ : BorelSpace E μ : Measure E inst✝ : IsAddHaarMeasure μ f : LocalHomeomorph E E hf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x g : E → F this : ↑f '' f.source = f.target x : E hx : x ∈ f.source ⊢ HasFDerivWithinAt (↑f) (f' x) f.source x ** exact (hf' x hx).hasFDerivWithinAt ** Qed
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Complex.circleTransformDeriv_periodic ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E ⊢ Periodic (circleTransformDeriv R z w f) (2 * π) ** have := periodic_circleMap ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E this : ∀ (c : ℂ) (R : ℝ), Periodic (circleMap c R) (2 * π) ⊢ Periodic (circleTransformDeriv R z w f) (2 * π) ** simp_rw [Periodic] at * ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E this : ∀ (c : ℂ) (R x : ℝ), circleMap c R (x + 2 * π) = circleMap c R x ⊢ ∀ (x : ℝ), circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x ** intro x ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E this : ∀ (c : ℂ) (R x : ℝ), circleMap c R (x + 2 * π) = circleMap c R x x : ℝ ⊢ circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x ** simp_rw [circleTransformDeriv, this] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E this : ∀ (c : ℂ) (R x : ℝ), circleMap c R (x + 2 * π) = circleMap c R x x : ℝ ⊢ (2 * ↑π * I)⁻¹ • deriv (circleMap z R) (x + 2 * π) • ((circleMap z R x - w) ^ 2)⁻¹ • f (circleMap z R x) = (2 * ↑π * I)⁻¹ • deriv (circleMap z R) x • ((circleMap z R x - w) ^ 2)⁻¹ • f (circleMap z R x) ** congr 2 ** case e_a.e_a E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E this : ∀ (c : ℂ) (R x : ℝ), circleMap c R (x + 2 * π) = circleMap c R x x : ℝ ⊢ deriv (circleMap z R) (x + 2 * π) = deriv (circleMap z R) x ** simp [this] ** Qed
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Complex.circleTransformDeriv_eq ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E ⊢ circleTransformDeriv R z w f = fun θ => (circleMap z R θ - w)⁻¹ • circleTransform R z w f θ ** ext ** case h E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E x✝ : ℝ ⊢ circleTransformDeriv R z w f x✝ = (circleMap z R x✝ - w)⁻¹ • circleTransform R z w f x✝ ** simp_rw [circleTransformDeriv, circleTransform, ← mul_smul, ← mul_assoc] ** case h E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E x✝ : ℝ ⊢ ((2 * ↑π * I)⁻¹ * deriv (circleMap z R) x✝ * ((circleMap z R x✝ - w) ^ 2)⁻¹) • f (circleMap z R x✝) = ((circleMap z R x✝ - w)⁻¹ * (2 * ↑π * I)⁻¹ * deriv (circleMap z R) x✝ * (circleMap z R x✝ - w)⁻¹) • f (circleMap z R x✝) ** ring_nf ** case h E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E x✝ : ℝ ⊢ ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * (-(circleMap z R x✝ * w * 2) + circleMap z R x✝ ^ 2 + w ^ 2)⁻¹ * (↑(Int.ofNat 1) / ↑2)) • f (circleMap z R x✝) = ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * (circleMap z R x✝ - w)⁻¹ ^ 2 * (↑(Int.ofNat 1) / ↑2)) • f (circleMap z R x✝) ** rw [inv_pow] ** case h E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E x✝ : ℝ ⊢ ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * (-(circleMap z R x✝ * w * 2) + circleMap z R x✝ ^ 2 + w ^ 2)⁻¹ * (↑(Int.ofNat 1) / ↑2)) • f (circleMap z R x✝) = ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * ((circleMap z R x✝ - w) ^ 2)⁻¹ * (↑(Int.ofNat 1) / ↑2)) • f (circleMap z R x✝) ** congr ** case h.e_a.e_a.e_a.e_a E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E x✝ : ℝ ⊢ -(circleMap z R x✝ * w * 2) + circleMap z R x✝ ^ 2 + w ^ 2 = (circleMap z R x✝ - w) ^ 2 ** ring ** Qed
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Complex.integral_circleTransform ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E ⊢ ∫ (θ : ℝ) in 0 ..2 * π, circleTransform R z w f θ = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(z, R), (z - w)⁻¹ • f z ** simp_rw [circleTransform, circleIntegral, deriv_circleMap, circleMap] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R : ℝ z w : ℂ f : ℂ → E ⊢ ∫ (θ : ℝ) in 0 ..2 * π, (2 * ↑π * I)⁻¹ • ((0 + ↑R * cexp (↑θ * I)) * I) • (z + ↑R * cexp (↑θ * I) - w)⁻¹ • f (z + ↑R * cexp (↑θ * I)) = (2 * ↑π * I)⁻¹ • ∫ (θ : ℝ) in 0 ..2 * π, ((0 + ↑R * cexp (↑θ * I)) * I) • (z + ↑R * cexp (↑θ * I) - w)⁻¹ • f (z + ↑R * cexp (↑θ * I)) ** simp ** Qed
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Complex.continuous_circleTransform ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous (circleTransform R z w f) ** apply_rules [Continuous.smul, continuous_const] ** case hg.hf E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => deriv (circleMap z R) x case hg.hg.hf E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => (circleMap z R x - w)⁻¹ case hg.hg.hg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => f (circleMap z R x) ** simp_rw [deriv_circleMap] ** case hg.hf E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => circleMap 0 R x * I case hg.hg.hf E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => (circleMap z R x - w)⁻¹ case hg.hg.hg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => f (circleMap z R x) ** apply_rules [Continuous.mul, continuous_circleMap 0 R, continuous_const] ** case hg.hg.hf E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => (circleMap z R x - w)⁻¹ ** apply continuous_circleMap_inv hw ** case hg.hg.hg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ Continuous fun x => f (circleMap z R x) ** apply ContinuousOn.comp_continuous hf (continuous_circleMap z R) ** case hg.hg.hg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w✝ : ℂ R : ℝ hR : 0 < R f : ℂ → E z w : ℂ hf : ContinuousOn f (sphere z R) hw : w ∈ ball z R ⊢ ∀ (x : ℝ), circleMap z R x ∈ sphere z R ** exact fun _ => (circleMap_mem_sphere _ hR.le) _ ** Qed
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Complex.continuousOn_prod_circle_transform_function ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z : ℂ ⊢ ContinuousOn (fun w => (circleMap z R w.2 - w.1)⁻¹ ^ 2) (closedBall z r ×ˢ univ) ** simp_rw [← one_div] ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z : ℂ ⊢ ContinuousOn (fun w => (1 / (circleMap z R w.2 - w.1)) ^ 2) (closedBall z r ×ˢ univ) ** apply_rules [ContinuousOn.pow, ContinuousOn.div, continuousOn_const] ** case hf.hg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z : ℂ ⊢ ContinuousOn (fun x => circleMap z R x.2 - x.1) (closedBall z r ×ˢ univ) case hf.h₀ E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z : ℂ ⊢ ∀ (x : ℂ × ℝ), x ∈ closedBall z r ×ˢ univ → circleMap z R x.2 - x.1 ≠ 0 ** refine' ((continuous_circleMap z R).continuousOn.comp continuousOn_snd fun _ => And.right).sub
(continuousOn_id.comp continuousOn_fst fun _ => And.left) ** case hf.h₀ E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z : ℂ ⊢ ∀ (x : ℂ × ℝ), x ∈ closedBall z r ×ˢ univ → circleMap z R x.2 - x.1 ≠ 0 ** simp only [mem_prod, Ne.def, and_imp, Prod.forall] ** case hf.h₀ E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z : ℂ ⊢ ∀ (a : ℂ) (b : ℝ), a ∈ closedBall z r → b ∈ univ → ¬circleMap z R b - a = 0 ** intro a b ha _ ** case hf.h₀ E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z a : ℂ b : ℝ ha : a ∈ closedBall z r a✝ : b ∈ univ ⊢ ¬circleMap z R b - a = 0 ** have ha2 : a ∈ ball z R := by simp at *; linarith ** case hf.h₀ E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z a : ℂ b : ℝ ha : a ∈ closedBall z r a✝ : b ∈ univ ha2 : a ∈ ball z R ⊢ ¬circleMap z R b - a = 0 ** exact sub_ne_zero.2 (circleMap_ne_mem_ball ha2 b) ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z a : ℂ b : ℝ ha : a ∈ closedBall z r a✝ : b ∈ univ ⊢ a ∈ ball z R ** simp at * ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E R✝ : ℝ z✝ w : ℂ R r : ℝ hr : r < R z a : ℂ b : ℝ ha : dist a z ≤ r a✝ : True ⊢ dist a z < R ** linarith ** Qed
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MeasureTheory.SimpleFunc.coe_injective ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f g : α →ₛ β H : ↑f = ↑g ⊢ f = g ** cases f ** case mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α g : α →ₛ β toFun✝ : α → β measurableSet_fiber'✝ : ∀ (x : β), MeasurableSet (toFun✝ ⁻¹' {x}) finite_range'✝ : Set.Finite (range toFun✝) H : ↑{ toFun := toFun✝, measurableSet_fiber' := measurableSet_fiber'✝, finite_range' := finite_range'✝ } = ↑g ⊢ { toFun := toFun✝, measurableSet_fiber' := measurableSet_fiber'✝, finite_range' := finite_range'✝ } = g ** cases g ** case mk.mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α toFun✝¹ : α → β measurableSet_fiber'✝¹ : ∀ (x : β), MeasurableSet (toFun✝¹ ⁻¹' {x}) finite_range'✝¹ : Set.Finite (range toFun✝¹) toFun✝ : α → β measurableSet_fiber'✝ : ∀ (x : β), MeasurableSet (toFun✝ ⁻¹' {x}) finite_range'✝ : Set.Finite (range toFun✝) H : ↑{ toFun := toFun✝¹, measurableSet_fiber' := measurableSet_fiber'✝¹, finite_range' := finite_range'✝¹ } = ↑{ toFun := toFun✝, measurableSet_fiber' := measurableSet_fiber'✝, finite_range' := finite_range'✝ } ⊢ { toFun := toFun✝¹, measurableSet_fiber' := measurableSet_fiber'✝¹, finite_range' := finite_range'✝¹ } = { toFun := toFun✝, measurableSet_fiber' := measurableSet_fiber'✝, finite_range' := finite_range'✝ } ** congr ** Qed
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MeasureTheory.SimpleFunc.forall_range_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f : α →ₛ β p : β → Prop ⊢ (∀ (y : β), y ∈ SimpleFunc.range f → p y) ↔ ∀ (x : α), p (↑f x) ** simp only [mem_range, Set.forall_range_iff] ** Qed
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MeasureTheory.SimpleFunc.exists_range_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f : α →ₛ β p : β → Prop ⊢ (∃ y, y ∈ SimpleFunc.range f ∧ p y) ↔ ∃ x, p (↑f x) ** simpa only [mem_range, exists_prop] using Set.exists_range_iff ** Qed
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MeasureTheory.SimpleFunc.range_const ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝² : MeasurableSpace α✝ α : Type u_5 inst✝¹ : MeasurableSpace α inst✝ : Nonempty α b : β ⊢ ↑(SimpleFunc.range (const α b)) = ↑{b} ** simp [Function.const] ** Qed
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MeasureTheory.SimpleFunc.range_const_subset ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α✝ α : Type u_5 inst✝ : MeasurableSpace α b : β ⊢ ↑(SimpleFunc.range (const α b)) ⊆ ↑{b} ** simp ** Qed
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MeasureTheory.SimpleFunc.measurableSet_cut ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α r : α → β → Prop f : α →ₛ β h : ∀ (b : β), MeasurableSet {a | r a b} ⊢ MeasurableSet {a | r a (↑f a)} ** have : { a | r a (f a) } = ⋃ b ∈ range f, { a | r a b } ∩ f ⁻¹' {b} := by
ext a
suffices r a (f a) ↔ ∃ i, r a (f i) ∧ f a = f i by simpa
exact ⟨fun h => ⟨a, ⟨h, rfl⟩⟩, fun ⟨a', ⟨h', e⟩⟩ => e.symm ▸ h'⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α r : α → β → Prop f : α →ₛ β h : ∀ (b : β), MeasurableSet {a | r a b} this : {a | r a (↑f a)} = ⋃ b ∈ range ↑f, {a | r a b} ∩ ↑f ⁻¹' {b} ⊢ MeasurableSet {a | r a (↑f a)} ** rw [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α r : α → β → Prop f : α →ₛ β h : ∀ (b : β), MeasurableSet {a | r a b} this : {a | r a (↑f a)} = ⋃ b ∈ range ↑f, {a | r a b} ∩ ↑f ⁻¹' {b} ⊢ MeasurableSet (⋃ b ∈ range ↑f, {a | r a b} ∩ ↑f ⁻¹' {b}) ** exact
MeasurableSet.biUnion f.finite_range.countable fun b _ =>
MeasurableSet.inter (h b) (f.measurableSet_fiber _) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α r : α → β → Prop f : α →ₛ β h : ∀ (b : β), MeasurableSet {a | r a b} ⊢ {a | r a (↑f a)} = ⋃ b ∈ range ↑f, {a | r a b} ∩ ↑f ⁻¹' {b} ** ext a ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α r : α → β → Prop f : α →ₛ β h : ∀ (b : β), MeasurableSet {a | r a b} a : α ⊢ a ∈ {a | r a (↑f a)} ↔ a ∈ ⋃ b ∈ range ↑f, {a | r a b} ∩ ↑f ⁻¹' {b} ** suffices r a (f a) ↔ ∃ i, r a (f i) ∧ f a = f i by simpa ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α r : α → β → Prop f : α →ₛ β h : ∀ (b : β), MeasurableSet {a | r a b} a : α ⊢ r a (↑f a) ↔ ∃ i, r a (↑f i) ∧ ↑f a = ↑f i ** exact ⟨fun h => ⟨a, ⟨h, rfl⟩⟩, fun ⟨a', ⟨h', e⟩⟩ => e.symm ▸ h'⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α r : α → β → Prop f : α →ₛ β h : ∀ (b : β), MeasurableSet {a | r a b} a : α this : r a (↑f a) ↔ ∃ i, r a (↑f i) ∧ ↑f a = ↑f i ⊢ a ∈ {a | r a (↑f a)} ↔ a ∈ ⋃ b ∈ range ↑f, {a | r a b} ∩ ↑f ⁻¹' {b} ** simpa ** Qed
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MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f : α →ₛ β μ : Measure α ⊢ ∑ y in SimpleFunc.range f, ↑↑μ (↑f ⁻¹' {y}) = ↑↑μ univ ** rw [f.sum_measure_preimage_singleton, coe_range, preimage_range] ** Qed
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MeasureTheory.SimpleFunc.piecewise_compl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α s : Set α hs : MeasurableSet sᶜ f g : α →ₛ β ⊢ ↑(piecewise sᶜ hs f g) = ↑(piecewise s (_ : MeasurableSet s) g f) ** simp [hs] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α s : Set α hs : MeasurableSet sᶜ f g : α →ₛ β ⊢ Set.piecewise sᶜ ↑f ↑g = Set.piecewise s ↑g ↑f ** convert Set.piecewise_compl s f g ** Qed
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MeasureTheory.SimpleFunc.range_indicator ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α s : Set α hs : MeasurableSet s hs_nonempty : Set.Nonempty s hs_ne_univ : s ≠ univ x y : β ⊢ SimpleFunc.range (piecewise s hs (const α x) (const α y)) = {x, y} ** simp only [← Finset.coe_inj, coe_range, coe_piecewise, range_piecewise, coe_const,
Finset.coe_insert, Finset.coe_singleton, hs_nonempty.image_const,
(nonempty_compl.2 hs_ne_univ).image_const, singleton_union, Function.const] ** Qed
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MeasureTheory.SimpleFunc.range_map ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α inst✝ : DecidableEq γ g : β → γ f : α →ₛ β ⊢ ↑(SimpleFunc.range (map g f)) = ↑(Finset.image g (SimpleFunc.range f)) ** simp only [coe_range, coe_map, Finset.coe_image, range_comp] ** Qed
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MeasureTheory.SimpleFunc.map_preimage ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f : α →ₛ β g : β → γ s : Set γ ⊢ ↑(map g f) ⁻¹' s = ↑f ⁻¹' ↑(Finset.filter (fun b => g b ∈ s) (SimpleFunc.range f)) ** simp only [coe_range, sep_mem_eq, coe_map, Finset.coe_filter,
← mem_preimage, inter_comm, preimage_inter_range, ← Finset.mem_coe] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f : α →ₛ β g : β → γ s : Set γ ⊢ g ∘ ↑f ⁻¹' s = ↑f ⁻¹' (g ⁻¹' s) ** exact preimage_comp ** Qed
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MeasureTheory.SimpleFunc.range_comp_subset_range ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β f : β →ₛ γ g : α → β hgm : Measurable g ⊢ ↑(SimpleFunc.range (comp f g hgm)) ⊆ ↑(SimpleFunc.range f) ** simp only [coe_range, coe_comp, Set.range_comp_subset_range] ** Qed
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