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MeasureTheory.SimpleFunc.bind_const ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f : α →ₛ β ⊢ bind f (const α) = f ** ext ** case H α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α f : α →ₛ β a✝ : α ⊢ ↑(bind f (const α)) a✝ = ↑f a✝ ** simp ** Qed
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MeasureTheory.SimpleFunc.range_one ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝² : MeasurableSpace α inst✝¹ : Nonempty α inst✝ : One β x : β ⊢ x ∈ SimpleFunc.range 1 ↔ x ∈ {1} ** simp [eq_comm] ** Qed
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MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α β : Type u_5 hα : IsEmpty α f : α →ₛ β ⊢ SimpleFunc.range f = ∅ ** rw [← Finset.not_nonempty_iff_eq_empty] ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α β : Type u_5 hα : IsEmpty α f : α →ₛ β ⊢ ¬Finset.Nonempty (SimpleFunc.range f) ** by_contra h ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α β : Type u_5 hα : IsEmpty α f : α →ₛ β h : Finset.Nonempty (SimpleFunc.range f) ⊢ False ** obtain ⟨y, hy_mem⟩ := h ** case intro α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α β : Type u_5 hα : IsEmpty α f : α →ₛ β y : β hy_mem : y ∈ SimpleFunc.range f ⊢ False ** rw [SimpleFunc.mem_range, Set.mem_range] at hy_mem ** case intro α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α β : Type u_5 hα : IsEmpty α f : α →ₛ β y : β hy_mem : ∃ y_1, ↑f y_1 = y ⊢ False ** obtain ⟨x, hxy⟩ := hy_mem ** case intro.intro α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α β : Type u_5 hα : IsEmpty α f : α →ₛ β y : β x : α hxy : ↑f x = y ⊢ False ** rw [isEmpty_iff] at hα ** case intro.intro α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α β : Type u_5 hα : α → False f : α →ₛ β y : β x : α hxy : ↑f x = y ⊢ False ** exact hα x ** Qed
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MeasureTheory.SimpleFunc.coe_restrict ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α K : Type u_5 inst✝ : Zero β f : α →ₛ β s : Set α hs : MeasurableSet s ⊢ ↑(restrict f s) = indicator s ↑f ** rw [restrict, dif_pos hs, coe_piecewise, coe_zero, piecewise_eq_indicator] ** Qed
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MeasureTheory.SimpleFunc.restrict_univ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α K : Type u_5 inst✝ : Zero β f : α →ₛ β ⊢ restrict f univ = f ** simp [restrict] ** Qed
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MeasureTheory.SimpleFunc.restrict_empty ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α K : Type u_5 inst✝ : Zero β f : α →ₛ β ⊢ restrict f ∅ = 0 ** simp [restrict] ** Qed
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MeasureTheory.SimpleFunc.map_restrict_of_zero ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝² : MeasurableSpace α K : Type u_5 inst✝¹ : Zero β inst✝ : Zero γ g : β → γ hg : g 0 = 0 f : α →ₛ β s : Set α x : α hs : MeasurableSet s ⊢ ↑(map g (restrict f s)) x = ↑(restrict (map g f) s) x ** simp [hs, Set.indicator_comp_of_zero hg] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝² : MeasurableSpace α K : Type u_5 inst✝¹ : Zero β inst✝ : Zero γ g : β → γ hg : g 0 = 0 f : α →ₛ β s : Set α x : α hs : ¬MeasurableSet s ⊢ ↑(map g (restrict f s)) x = ↑(restrict (map g f) s) x ** simp [restrict_of_not_measurable hs, hg] ** Qed
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MeasureTheory.SimpleFunc.restrict_apply ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α K : Type u_5 inst✝ : Zero β f : α →ₛ β s : Set α hs : MeasurableSet s a : α ⊢ ↑(restrict f s) a = indicator s (↑f) a ** simp only [f.coe_restrict hs] ** Qed
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MeasureTheory.SimpleFunc.restrict_preimage ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α K : Type u_5 inst✝ : Zero β f : α →ₛ β s : Set α hs : MeasurableSet s t : Set β ht : ¬0 ∈ t ⊢ ↑(restrict f s) ⁻¹' t = s ∩ ↑f ⁻¹' t ** simp [hs, indicator_preimage_of_not_mem _ _ ht, inter_comm] ** Qed
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MeasureTheory.SimpleFunc.mem_restrict_range ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α K : Type u_5 inst✝ : Zero β r : β s : Set α f : α →ₛ β hs : MeasurableSet s ⊢ r ∈ SimpleFunc.range (restrict f s) ↔ r = 0 ∧ s ≠ univ ∨ r ∈ ↑f '' s ** rw [← Finset.mem_coe, coe_range, coe_restrict _ hs, mem_range_indicator] ** Qed
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MeasureTheory.SimpleFunc.restrict_mono ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝² : MeasurableSpace α K : Type u_5 inst✝¹ : Zero β inst✝ : Preorder β s : Set α f g : α →ₛ β H : f ≤ g hs : MeasurableSet s x : α ⊢ ↑(restrict f s) x ≤ ↑(restrict g s) x ** simp only [coe_restrict _ hs, indicator_le_indicator (H x)] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝² : MeasurableSpace α K : Type u_5 inst✝¹ : Zero β inst✝ : Preorder β s : Set α f g : α →ₛ β H : f ≤ g hs : ¬MeasurableSet s ⊢ restrict f s ≤ restrict g s ** simp only [restrict_of_not_measurable hs, le_refl] ** Qed
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MeasureTheory.SimpleFunc.approx_apply ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁷ : MeasurableSpace α K : Type u_5 inst✝⁶ : SemilatticeSup β inst✝⁵ : OrderBot β inst✝⁴ : Zero β inst✝³ : TopologicalSpace β inst✝² : OrderClosedTopology β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β n : ℕ a : α hf : Measurable f ⊢ ↑(approx i f n) a = Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0 ** dsimp only [approx] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁷ : MeasurableSpace α K : Type u_5 inst✝⁶ : SemilatticeSup β inst✝⁵ : OrderBot β inst✝⁴ : Zero β inst✝³ : TopologicalSpace β inst✝² : OrderClosedTopology β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β n : ℕ a : α hf : Measurable f ⊢ ↑(Finset.sup (Finset.range n) fun k => restrict (const α (i k)) {a | i k ≤ f a}) a = Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0 ** rw [finset_sup_apply] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁷ : MeasurableSpace α K : Type u_5 inst✝⁶ : SemilatticeSup β inst✝⁵ : OrderBot β inst✝⁴ : Zero β inst✝³ : TopologicalSpace β inst✝² : OrderClosedTopology β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β n : ℕ a : α hf : Measurable f ⊢ (Finset.sup (Finset.range n) fun c => ↑(restrict (const α (i c)) {a | i c ≤ f a}) a) = Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else 0 ** congr ** case e_f α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁷ : MeasurableSpace α K : Type u_5 inst✝⁶ : SemilatticeSup β inst✝⁵ : OrderBot β inst✝⁴ : Zero β inst✝³ : TopologicalSpace β inst✝² : OrderClosedTopology β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β n : ℕ a : α hf : Measurable f ⊢ (fun c => ↑(restrict (const α (i c)) {a | i c ≤ f a}) a) = fun k => if i k ≤ f a then i k else 0 ** funext k ** case e_f.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁷ : MeasurableSpace α K : Type u_5 inst✝⁶ : SemilatticeSup β inst✝⁵ : OrderBot β inst✝⁴ : Zero β inst✝³ : TopologicalSpace β inst✝² : OrderClosedTopology β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β n : ℕ a : α hf : Measurable f k : ℕ ⊢ ↑(restrict (const α (i k)) {a | i k ≤ f a}) a = if i k ≤ f a then i k else 0 ** rw [restrict_apply] ** case e_f.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁷ : MeasurableSpace α K : Type u_5 inst✝⁶ : SemilatticeSup β inst✝⁵ : OrderBot β inst✝⁴ : Zero β inst✝³ : TopologicalSpace β inst✝² : OrderClosedTopology β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β n : ℕ a : α hf : Measurable f k : ℕ ⊢ indicator {a | i k ≤ f a} (↑(const α (i k))) a = if i k ≤ f a then i k else 0 ** simp only [coe_const, mem_setOf_eq, indicator_apply, Function.const_apply] ** case e_f.h.hs α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁷ : MeasurableSpace α K : Type u_5 inst✝⁶ : SemilatticeSup β inst✝⁵ : OrderBot β inst✝⁴ : Zero β inst✝³ : TopologicalSpace β inst✝² : OrderClosedTopology β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β n : ℕ a : α hf : Measurable f k : ℕ ⊢ MeasurableSet {a | i k ≤ f a} ** exact hf measurableSet_Ici ** Qed
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MeasureTheory.SimpleFunc.approx_comp ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁸ : MeasurableSpace α K : Type u_5 inst✝⁷ : SemilatticeSup β inst✝⁶ : OrderBot β inst✝⁵ : Zero β inst✝⁴ : TopologicalSpace β inst✝³ : OrderClosedTopology β inst✝² : MeasurableSpace β inst✝¹ : OpensMeasurableSpace β inst✝ : MeasurableSpace γ i : ℕ → β f : γ → β g : α → γ n : ℕ a : α hf : Measurable f hg : Measurable g ⊢ ↑(approx i (f ∘ g) n) a = ↑(approx i f n) (g a) ** rw [approx_apply _ hf, approx_apply _ (hf.comp hg), Function.comp_apply] ** Qed
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MeasureTheory.SimpleFunc.iSup_approx_apply ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ ⊢ ⨆ n, ↑(approx i f n) a = ⨆ k, ⨆ (_ : i k ≤ f a), i k ** refine' le_antisymm (iSup_le fun n => _) (iSup_le fun k => iSup_le fun hk => _) ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ n : ℕ ⊢ ↑(approx i f n) a ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k ** rw [approx_apply a hf, h_zero] ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ n : ℕ ⊢ (Finset.sup (Finset.range n) fun k => if i k ≤ f a then i k else ⊥) ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k ** refine' Finset.sup_le fun k _ => _ ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ n k : ℕ x✝ : k ∈ Finset.range n ⊢ (if i k ≤ f a then i k else ⊥) ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k ** split_ifs with h ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ n k : ℕ x✝ : k ∈ Finset.range n h : i k ≤ f a ⊢ i k ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k ** exact le_iSup_of_le k (le_iSup (fun _ : i k ≤ f a => i k) h) ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ n k : ℕ x✝ : k ∈ Finset.range n h : ¬i k ≤ f a ⊢ ⊥ ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k ** exact bot_le ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ k : ℕ hk : i k ≤ f a ⊢ i k ≤ ⨆ n, ↑(approx i f n) a ** refine' le_iSup_of_le (k + 1) _ ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ k : ℕ hk : i k ≤ f a ⊢ i k ≤ ↑(approx i f (k + 1)) a ** rw [approx_apply a hf] ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ k : ℕ hk : i k ≤ f a ⊢ i k ≤ Finset.sup (Finset.range (k + 1)) fun k => if i k ≤ f a then i k else 0 ** have : k ∈ Finset.range (k + 1) := Finset.mem_range.2 (Nat.lt_succ_self _) ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ k : ℕ hk : i k ≤ f a this : k ∈ Finset.range (k + 1) ⊢ i k ≤ Finset.sup (Finset.range (k + 1)) fun k => if i k ≤ f a then i k else 0 ** refine' le_trans (le_of_eq _) (Finset.le_sup this) ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁶ : MeasurableSpace α K : Type u_5 inst✝⁵ : TopologicalSpace β inst✝⁴ : CompleteLattice β inst✝³ : OrderClosedTopology β inst✝² : Zero β inst✝¹ : MeasurableSpace β inst✝ : OpensMeasurableSpace β i : ℕ → β f : α → β a : α hf : Measurable f h_zero : 0 = ⊥ k : ℕ hk : i k ≤ f a this : k ∈ Finset.range (k + 1) ⊢ i k = if i k ≤ f a then i k else 0 ** rw [if_pos hk] ** Qed
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MeasureTheory.SimpleFunc.ennrealRatEmbed_encode ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 q : ℚ ⊢ ennrealRatEmbed (Encodable.encode q) = ↑(Real.toNNReal ↑q) ** rw [ennrealRatEmbed, Encodable.encodek] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 q : ℚ ⊢ ENNReal.ofReal ↑(Option.getD (Option.some q) 0) = ↑(Real.toNNReal ↑q) ** rfl ** Qed
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MeasureTheory.SimpleFunc.eapprox_lt_top ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 f : α → ℝ≥0∞ n : ℕ a : α ⊢ ↑(eapprox f n) a < ⊤ ** simp only [eapprox, approx, finset_sup_apply, Finset.mem_range, ENNReal.bot_eq_zero, restrict] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 f : α → ℝ≥0∞ n : ℕ a : α ⊢ (Finset.sup (Finset.range n) fun c => ↑(if hs : MeasurableSet {a | ennrealRatEmbed c ≤ f a} then piecewise {a | ennrealRatEmbed c ≤ f a} hs (const α (ennrealRatEmbed c)) 0 else 0) a) < ⊤ ** rw [Finset.sup_lt_iff (α := ℝ≥0∞) WithTop.zero_lt_top] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 f : α → ℝ≥0∞ n : ℕ a : α ⊢ ∀ (b : ℕ), b ∈ Finset.range n → ↑(if hs : MeasurableSet {a | ennrealRatEmbed b ≤ f a} then piecewise {a | ennrealRatEmbed b ≤ f a} hs (const α (ennrealRatEmbed b)) 0 else 0) a < ⊤ ** intro b _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 f : α → ℝ≥0∞ n : ℕ a : α b : ℕ a✝ : b ∈ Finset.range n ⊢ ↑(if hs : MeasurableSet {a | ennrealRatEmbed b ≤ f a} then piecewise {a | ennrealRatEmbed b ≤ f a} hs (const α (ennrealRatEmbed b)) 0 else 0) a < ⊤ ** split_ifs ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 f : α → ℝ≥0∞ n : ℕ a : α b : ℕ a✝ : b ∈ Finset.range n h✝ : MeasurableSet {a | ennrealRatEmbed b ≤ f a} ⊢ ↑(piecewise {a | ennrealRatEmbed b ≤ f a} h✝ (const α (ennrealRatEmbed b)) 0) a < ⊤ ** simp only [coe_zero, coe_piecewise, piecewise_eq_indicator, coe_const] ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 f : α → ℝ≥0∞ n : ℕ a : α b : ℕ a✝ : b ∈ Finset.range n h✝ : MeasurableSet {a | ennrealRatEmbed b ≤ f a} ⊢ Set.piecewise {a | ennrealRatEmbed b ≤ f a} (Function.const α (ennrealRatEmbed b)) 0 a < ⊤ ** calc
{ a : α | ennrealRatEmbed b ≤ f a }.indicator (fun _ => ennrealRatEmbed b) a ≤
ennrealRatEmbed b :=
indicator_le_self _ _ a
_ < ⊤ := ENNReal.coe_lt_top ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝ : MeasurableSpace α K : Type u_5 f : α → ℝ≥0∞ n : ℕ a : α b : ℕ a✝ : b ∈ Finset.range n h✝ : ¬MeasurableSet {a | ennrealRatEmbed b ≤ f a} ⊢ ↑0 a < ⊤ ** exact WithTop.zero_lt_top ** Qed
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MeasureTheory.SimpleFunc.map_lintegral ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β ⊢ lintegral (map g f) μ = ∑ x in SimpleFunc.range f, g x * ↑↑μ (↑f ⁻¹' {x}) ** simp only [lintegral, range_map] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β ⊢ ∑ x in Finset.image g (SimpleFunc.range f), x * ↑↑μ (↑(map g f) ⁻¹' {x}) = ∑ x in SimpleFunc.range f, g x * ↑↑μ (↑f ⁻¹' {x}) ** refine' Finset.sum_image' _ fun b hb => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β b : β hb : b ∈ SimpleFunc.range f ⊢ g b * ↑↑μ (↑(map g f) ⁻¹' {g b}) = ∑ x in Finset.filter (fun c' => g c' = g b) (SimpleFunc.range f), g x * ↑↑μ (↑f ⁻¹' {x}) ** rcases mem_range.1 hb with ⟨a, rfl⟩ ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β a : α hb : ↑f a ∈ SimpleFunc.range f ⊢ g (↑f a) * ↑↑μ (↑(map g f) ⁻¹' {g (↑f a)}) = ∑ x in Finset.filter (fun c' => g c' = g (↑f a)) (SimpleFunc.range f), g x * ↑↑μ (↑f ⁻¹' {x}) ** rw [map_preimage_singleton, ← f.sum_measure_preimage_singleton, Finset.mul_sum] ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β a : α hb : ↑f a ∈ SimpleFunc.range f ⊢ ∑ x in Finset.filter (fun b => g b = g (↑f a)) (SimpleFunc.range f), g (↑f a) * ↑↑μ (↑f ⁻¹' {x}) = ∑ x in Finset.filter (fun c' => g c' = g (↑f a)) (SimpleFunc.range f), g x * ↑↑μ (↑f ⁻¹' {x}) ** refine' Finset.sum_congr _ _ ** case intro.refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β a : α hb : ↑f a ∈ SimpleFunc.range f ⊢ Finset.filter (fun b => g b = g (↑f a)) (SimpleFunc.range f) = Finset.filter (fun c' => g c' = g (↑f a)) (SimpleFunc.range f) ** congr ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β a : α hb : ↑f a ∈ SimpleFunc.range f ⊢ ∀ (x : β), x ∈ Finset.filter (fun c' => g c' = g (↑f a)) (SimpleFunc.range f) → g (↑f a) * ↑↑μ (↑f ⁻¹' {x}) = g x * ↑↑μ (↑f ⁻¹' {x}) ** intro x ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β a : α hb : ↑f a ∈ SimpleFunc.range f x : β ⊢ x ∈ Finset.filter (fun c' => g c' = g (↑f a)) (SimpleFunc.range f) → g (↑f a) * ↑↑μ (↑f ⁻¹' {x}) = g x * ↑↑μ (↑f ⁻¹' {x}) ** simp only [Finset.mem_filter] ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β a : α hb : ↑f a ∈ SimpleFunc.range f x : β ⊢ x ∈ SimpleFunc.range f ∧ g x = g (↑f a) → g (↑f a) * ↑↑μ (↑f ⁻¹' {x}) = g x * ↑↑μ (↑f ⁻¹' {x}) ** rintro ⟨_, h⟩ ** case intro.refine'_2.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α g : β → ℝ≥0∞ f : α →ₛ β a : α hb : ↑f a ∈ SimpleFunc.range f x : β left✝ : x ∈ SimpleFunc.range f h : g x = g (↑f a) ⊢ g (↑f a) * ↑↑μ (↑f ⁻¹' {x}) = g x * ↑↑μ (↑f ⁻¹' {x}) ** rw [h] ** Qed
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MeasureTheory.SimpleFunc.add_lintegral ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α →ₛ ℝ≥0∞ ⊢ lintegral (f + g) μ = ∑ x in SimpleFunc.range (pair f g), (x.1 * ↑↑μ (↑(pair f g) ⁻¹' {x}) + x.2 * ↑↑μ (↑(pair f g) ⁻¹' {x})) ** rw [add_eq_map₂, map_lintegral] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α →ₛ ℝ≥0∞ ⊢ ∑ x in SimpleFunc.range (pair f g), (x.1 + x.2) * ↑↑μ (↑(pair f g) ⁻¹' {x}) = ∑ x in SimpleFunc.range (pair f g), (x.1 * ↑↑μ (↑(pair f g) ⁻¹' {x}) + x.2 * ↑↑μ (↑(pair f g) ⁻¹' {x})) ** exact Finset.sum_congr rfl fun a _ => add_mul _ _ _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α →ₛ ℝ≥0∞ ⊢ ∑ x in SimpleFunc.range (pair f g), (x.1 * ↑↑μ (↑(pair f g) ⁻¹' {x}) + x.2 * ↑↑μ (↑(pair f g) ⁻¹' {x})) = ∑ x in SimpleFunc.range (pair f g), x.1 * ↑↑μ (↑(pair f g) ⁻¹' {x}) + ∑ x in SimpleFunc.range (pair f g), x.2 * ↑↑μ (↑(pair f g) ⁻¹' {x}) ** rw [Finset.sum_add_distrib] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α →ₛ ℝ≥0∞ ⊢ ∑ x in SimpleFunc.range (pair f g), x.1 * ↑↑μ (↑(pair f g) ⁻¹' {x}) + ∑ x in SimpleFunc.range (pair f g), x.2 * ↑↑μ (↑(pair f g) ⁻¹' {x}) = lintegral (map Prod.fst (pair f g)) μ + lintegral (map Prod.snd (pair f g)) μ ** rw [map_lintegral, map_lintegral] ** Qed
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MeasureTheory.SimpleFunc.lintegral_sum ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m✝ : MeasurableSpace α μ✝ ν : Measure α m : MeasurableSpace α ι : Type u_5 f : α →ₛ ℝ≥0∞ μ : ι → Measure α ⊢ lintegral f (Measure.sum μ) = ∑' (i : ι), lintegral f (μ i) ** simp only [lintegral, Measure.sum_apply, f.measurableSet_preimage, ← Finset.tsum_subtype, ←
ENNReal.tsum_mul_left] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m✝ : MeasurableSpace α μ✝ ν : Measure α m : MeasurableSpace α ι : Type u_5 f : α →ₛ ℝ≥0∞ μ : ι → Measure α ⊢ ∑' (x : { x // x ∈ SimpleFunc.range f }) (i : ι), ↑x * ↑↑(μ i) (↑f ⁻¹' {↑x}) = ∑' (i : ι) (x : { x // x ∈ SimpleFunc.range f }), ↑x * ↑↑(μ i) (↑f ⁻¹' {↑x}) ** apply ENNReal.tsum_comm ** Qed
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MeasureTheory.SimpleFunc.restrict_lintegral ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α →ₛ ℝ≥0∞ s : Set α hs : MeasurableSet s x : α hx : ↑(restrict f s) x ≠ 0 hxs : x ∈ s x✝ : ↑↑μ (↑(restrict f s) ⁻¹' {↑(restrict f s) x}) ≠ 0 ⊢ ↑(restrict f s) x ∈ SimpleFunc.range f ** simp only [f.restrict_apply hs, indicator_of_mem hxs, mem_range_self] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α →ₛ ℝ≥0∞ s : Set α hs : MeasurableSet s x : α hx : ↑(restrict f s) x ≠ 0 hxs : ¬x ∈ s ⊢ ↑(restrict f s) x = 0 ** simp [*] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α →ₛ ℝ≥0∞ s : Set α hs : MeasurableSet s b : α hb : ↑f b = 0 ⊢ ↑f b * ↑↑μ (↑(restrict f s) ⁻¹' {↑f b}) = ↑f b * ↑↑μ (↑f ⁻¹' {↑f b} ∩ s) ** simp only [hb, zero_mul] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α →ₛ ℝ≥0∞ s : Set α hs : MeasurableSet s b : α hb : ¬↑f b = 0 ⊢ ↑f b * ↑↑μ (↑(restrict f s) ⁻¹' {↑f b}) = ↑f b * ↑↑μ (↑f ⁻¹' {↑f b} ∩ s) ** rw [restrict_preimage_singleton _ hs hb, inter_comm] ** Qed
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MeasureTheory.SimpleFunc.lintegral_restrict ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m✝ : MeasurableSpace α μ✝ ν : Measure α m : MeasurableSpace α f : α →ₛ ℝ≥0∞ s : Set α μ : Measure α ⊢ lintegral f (Measure.restrict μ s) = ∑ y in SimpleFunc.range f, y * ↑↑μ (↑f ⁻¹' {y} ∩ s) ** simp only [lintegral, Measure.restrict_apply, f.measurableSet_preimage] ** Qed
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MeasureTheory.SimpleFunc.const_lintegral ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α c : ℝ≥0∞ ⊢ lintegral (const α c) μ = c * ↑↑μ univ ** rw [lintegral] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α c : ℝ≥0∞ ⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ ** cases isEmpty_or_nonempty α ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α c : ℝ≥0∞ h✝ : IsEmpty α ⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ ** simp [μ.eq_zero_of_isEmpty] ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α c : ℝ≥0∞ h✝ : Nonempty α ⊢ ∑ x in SimpleFunc.range (const α c), x * ↑↑μ (↑(const α c) ⁻¹' {x}) = c * ↑↑μ univ ** simp ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α c : ℝ≥0∞ h✝ : Nonempty α ⊢ c * ↑↑μ (Function.const α c ⁻¹' {c}) = c * ↑↑μ univ ** unfold Function.const ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α c : ℝ≥0∞ h✝ : Nonempty α ⊢ c * ↑↑μ ((fun x => c) ⁻¹' {c}) = c * ↑↑μ univ ** rw [preimage_const_of_mem (mem_singleton c)] ** Qed
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MeasureTheory.SimpleFunc.const_lintegral_restrict ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α c : ℝ≥0∞ s : Set α ⊢ lintegral (const α c) (Measure.restrict μ s) = c * ↑↑μ s ** rw [const_lintegral, Measure.restrict_apply MeasurableSet.univ, univ_inter] ** Qed
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MeasureTheory.SimpleFunc.lintegral_congr ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α →ₛ ℝ≥0∞ h : ↑f =ᵐ[μ] ↑g y : ℝ≥0∞ x : α hx : ↑f x = ↑g x ⊢ x ∈ ↑f ⁻¹' {y} ↔ x ∈ ↑g ⁻¹' {y} ** simp [hx] ** Qed
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MeasureTheory.SimpleFunc.lintegral_map' ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α β : Type u_5 inst✝ : MeasurableSpace β μ' : Measure β f : α →ₛ ℝ≥0∞ g : β →ₛ ℝ≥0∞ m' : α → β eq : ∀ (a : α), ↑f a = ↑g (m' a) h : ∀ (s : Set β), MeasurableSet s → ↑↑μ' s = ↑↑μ (m' ⁻¹' s) y : ℝ≥0∞ ⊢ ↑↑μ (↑f ⁻¹' {y}) = ↑↑μ' (↑g ⁻¹' {y}) ** simp only [preimage, eq] ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α β : Type u_5 inst✝ : MeasurableSpace β μ' : Measure β f : α →ₛ ℝ≥0∞ g : β →ₛ ℝ≥0∞ m' : α → β eq : ∀ (a : α), ↑f a = ↑g (m' a) h : ∀ (s : Set β), MeasurableSet s → ↑↑μ' s = ↑↑μ (m' ⁻¹' s) y : ℝ≥0∞ ⊢ ↑↑μ {x | ↑g (m' x) ∈ {y}} = ↑↑μ' {x | ↑g x ∈ {y}} ** exact (h (g ⁻¹' {y}) (g.measurableSet_preimage _)).symm ** Qed
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MeasureTheory.SimpleFunc.support_eq ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹ : MeasurableSpace α inst✝ : Zero β f : α →ₛ β x : α ⊢ x ∈ support ↑f ↔ x ∈ ⋃ y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f), ↑f ⁻¹' {y} ** simp only [mem_support, Set.mem_preimage, mem_filter, mem_range_self, true_and_iff, exists_prop,
mem_iUnion, Set.mem_range, mem_singleton_iff, exists_eq_right'] ** Qed
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MeasureTheory.SimpleFunc.measurableSet_support ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝² : Zero β inst✝¹ : Zero γ μ : Measure α f✝ : α →ₛ β inst✝ : MeasurableSpace α f : α →ₛ β ⊢ MeasurableSet (support ↑f) ** rw [f.support_eq] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝² : Zero β inst✝¹ : Zero γ μ : Measure α f✝ : α →ₛ β inst✝ : MeasurableSpace α f : α →ₛ β ⊢ MeasurableSet (⋃ y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f), ↑f ⁻¹' {y}) ** exact Finset.measurableSet_biUnion _ fun y _ => measurableSet_fiber _ _ ** Qed
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MeasureTheory.SimpleFunc.finMeasSupp_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f : α →ₛ β ⊢ SimpleFunc.FinMeasSupp f μ ↔ ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ** constructor ** case mp α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f : α →ₛ β ⊢ SimpleFunc.FinMeasSupp f μ → ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ** refine' fun h y hy => lt_of_le_of_lt (measure_mono _) h ** case mp α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f : α →ₛ β h : SimpleFunc.FinMeasSupp f μ y : β hy : y ≠ 0 ⊢ ↑f ⁻¹' {y} ⊆ {x | (fun x => ↑f x = OfNat.ofNat 0 x) x}ᶜ ** exact fun x hx (H : f x = 0) => hy <| H ▸ Eq.symm hx ** case mpr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f : α →ₛ β ⊢ (∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤) → SimpleFunc.FinMeasSupp f μ ** intro H ** case mpr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f : α →ₛ β H : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ⊢ SimpleFunc.FinMeasSupp f μ ** rw [finMeasSupp_iff_support, support_eq] ** case mpr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f : α →ₛ β H : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ⊢ ↑↑μ (⋃ y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f), ↑f ⁻¹' {y}) < ⊤ ** refine' lt_of_le_of_lt (measure_biUnion_finset_le _ _) (sum_lt_top _) ** case mpr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f : α →ₛ β H : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ⊢ ∀ (a : β), a ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f) → ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** exact fun y hy => (H y (Finset.mem_filter.1 hy).2).ne ** Qed
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MeasureTheory.SimpleFunc.FinMeasSupp.add ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝² : Zero β✝ inst✝¹ : Zero γ μ : Measure α f✝ : α →ₛ β✝ β : Type u_5 inst✝ : AddMonoid β f g : α →ₛ β hf : SimpleFunc.FinMeasSupp f μ hg : SimpleFunc.FinMeasSupp g μ ⊢ SimpleFunc.FinMeasSupp (f + g) μ ** rw [add_eq_map₂] ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝² : Zero β✝ inst✝¹ : Zero γ μ : Measure α f✝ : α →ₛ β✝ β : Type u_5 inst✝ : AddMonoid β f g : α →ₛ β hf : SimpleFunc.FinMeasSupp f μ hg : SimpleFunc.FinMeasSupp g μ ⊢ SimpleFunc.FinMeasSupp (map (fun p => p.1 + p.2) (pair f g)) μ ** exact hf.map₂ hg (zero_add 0) ** Qed
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MeasureTheory.SimpleFunc.FinMeasSupp.lintegral_lt_top ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤ ⊢ lintegral f μ < ⊤ ** refine' sum_lt_top fun a ha => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤ a : ℝ≥0∞ ha : a ∈ SimpleFunc.range f ⊢ a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** rcases eq_or_ne a ∞ with (rfl | ha) ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤ ha : ⊤ ∈ SimpleFunc.range f ⊢ ⊤ * ↑↑μ (↑f ⁻¹' {⊤}) ≠ ⊤ ** simp only [ae_iff, Ne.def, Classical.not_not] at hf ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ ha : ⊤ ∈ SimpleFunc.range f hf : ↑↑μ {a | ↑f a = ⊤} = 0 ⊢ ⊤ * ↑↑μ (↑f ⁻¹' {⊤}) ≠ ⊤ ** simp [Set.preimage, hf] ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤ a : ℝ≥0∞ ha✝ : a ∈ SimpleFunc.range f ha : a ≠ ⊤ ⊢ a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** by_cases ha0 : a = 0 ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤ a : ℝ≥0∞ ha✝ : a ∈ SimpleFunc.range f ha : a ≠ ⊤ ha0 : a = 0 ⊢ a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** subst a ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤ ha✝ : 0 ∈ SimpleFunc.range f ha : 0 ≠ ⊤ ⊢ 0 * ↑↑μ (↑f ⁻¹' {0}) ≠ ⊤ ** rwa [zero_mul] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ hm : SimpleFunc.FinMeasSupp f μ hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤ a : ℝ≥0∞ ha✝ : a ∈ SimpleFunc.range f ha : a ≠ ⊤ ha0 : ¬a = 0 ⊢ a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** exact mul_ne_top ha (finMeasSupp_iff.1 hm _ ha0).ne ** Qed
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MeasureTheory.SimpleFunc.FinMeasSupp.of_lintegral_ne_top ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ h : lintegral f μ ≠ ⊤ ⊢ SimpleFunc.FinMeasSupp f μ ** refine' finMeasSupp_iff.2 fun b hb => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ h : lintegral f μ ≠ ⊤ b : ℝ≥0∞ hb : b ≠ 0 ⊢ ↑↑μ (↑f ⁻¹' {b}) < ⊤ ** rw [f.lintegral_eq_of_subset' (Finset.subset_insert b _)] at h ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ b : ℝ≥0∞ h : ∑ x in insert b (SimpleFunc.range f \ {0}), x * ↑↑μ (↑f ⁻¹' {x}) ≠ ⊤ hb : b ≠ 0 ⊢ ↑↑μ (↑f ⁻¹' {b}) < ⊤ ** refine' ENNReal.lt_top_of_mul_ne_top_right _ hb ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α inst✝¹ : Zero β inst✝ : Zero γ μ : Measure α f✝ : α →ₛ β f : α →ₛ ℝ≥0∞ b : ℝ≥0∞ h : ∑ x in insert b (SimpleFunc.range f \ {0}), x * ↑↑μ (↑f ⁻¹' {x}) ≠ ⊤ hb : b ≠ 0 ⊢ b * ↑↑μ (↑f ⁻¹' {b}) ≠ ⊤ ** exact (lt_top_of_sum_ne_top h (Finset.mem_insert_self _ _)).ne ** Qed
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Measurable.add_simpleFunc ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f : α →ₛ E ⊢ Measurable (g + ↑f) ** induction' f using SimpleFunc.induction with c s hs f f' hff' hf hf' ** case h_ind α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g c : E s : Set α hs : MeasurableSet s ⊢ Measurable (g + ↑(piecewise s hs (const α c) (const α 0))) ** simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
SimpleFunc.coe_zero] ** case h_ind α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g c : E s : Set α hs : MeasurableSet s ⊢ Measurable (g + Set.piecewise s (Function.const α c) 0) ** rw [← piecewise_same s g, ← piecewise_add] ** case h_ind α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g c : E s : Set α hs : MeasurableSet s ⊢ Measurable (Set.piecewise s (g + Function.const α c) (g + 0)) ** exact Measurable.piecewise hs (hg.add_const _) (hg.add_const _) ** case h_add α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (g + ↑f) hf' : Measurable (g + ↑f') this : g + ↑(f + f') = Set.piecewise (support ↑f) (g + ↑f) (g + ↑f') ⊢ Measurable (g + ↑(f + f')) ** rw [this] ** case h_add α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (g + ↑f) hf' : Measurable (g + ↑f') this : g + ↑(f + f') = Set.piecewise (support ↑f) (g + ↑f) (g + ↑f') ⊢ Measurable (Set.piecewise (support ↑f) (g + ↑f) (g + ↑f')) ** exact Measurable.piecewise f.measurableSet_support hf hf' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (g + ↑f) hf' : Measurable (g + ↑f') ⊢ g + ↑(f + f') = Set.piecewise (support ↑f) (g + ↑f) (g + ↑f') ** ext x ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (g + ↑f) hf' : Measurable (g + ↑f') x : α ⊢ (g + ↑(f + f')) x = Set.piecewise (support ↑f) (g + ↑f) (g + ↑f') x ** by_cases hx : x ∈ Function.support f ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (g + ↑f) hf' : Measurable (g + ↑f') x : α hx : x ∈ support ↑f ⊢ (g + ↑(f + f')) x = Set.piecewise (support ↑f) (g + ↑f) (g + ↑f') x ** simpa only [SimpleFunc.coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_mem _ _ _ hx, _root_.add_right_inj, add_right_eq_self]
using Set.disjoint_left.1 hff' hx ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (g + ↑f) hf' : Measurable (g + ↑f') x : α hx : ¬x ∈ support ↑f ⊢ (g + ↑(f + f')) x = Set.piecewise (support ↑f) (g + ↑f) (g + ↑f') x ** simpa only [SimpleFunc.coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_not_mem _ _ _ hx, _root_.add_right_inj, add_left_eq_self] using hx ** Qed
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Measurable.simpleFunc_add ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f : α →ₛ E ⊢ Measurable (↑f + g) ** induction' f using SimpleFunc.induction with c s hs f f' hff' hf hf' ** case h_ind α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g c : E s : Set α hs : MeasurableSet s ⊢ Measurable (↑(piecewise s hs (const α c) (const α 0)) + g) ** simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
SimpleFunc.coe_zero] ** case h_ind α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g c : E s : Set α hs : MeasurableSet s ⊢ Measurable (Set.piecewise s (Function.const α c) 0 + g) ** rw [← piecewise_same s g, ← piecewise_add] ** case h_ind α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g c : E s : Set α hs : MeasurableSet s ⊢ Measurable (Set.piecewise s (Function.const α c + g) (0 + g)) ** exact Measurable.piecewise hs (hg.const_add _) (hg.const_add _) ** case h_add α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (↑f + g) hf' : Measurable (↑f' + g) this : ↑(f + f') + g = Set.piecewise (support ↑f) (↑f + g) (↑f' + g) ⊢ Measurable (↑(f + f') + g) ** rw [this] ** case h_add α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (↑f + g) hf' : Measurable (↑f' + g) this : ↑(f + f') + g = Set.piecewise (support ↑f) (↑f + g) (↑f' + g) ⊢ Measurable (Set.piecewise (support ↑f) (↑f + g) (↑f' + g)) ** exact Measurable.piecewise f.measurableSet_support hf hf' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (↑f + g) hf' : Measurable (↑f' + g) ⊢ ↑(f + f') + g = Set.piecewise (support ↑f) (↑f + g) (↑f' + g) ** ext x ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (↑f + g) hf' : Measurable (↑f' + g) x : α ⊢ (↑(f + f') + g) x = Set.piecewise (support ↑f) (↑f + g) (↑f' + g) x ** by_cases hx : x ∈ Function.support f ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (↑f + g) hf' : Measurable (↑f' + g) x : α hx : x ∈ support ↑f ⊢ (↑(f + f') + g) x = Set.piecewise (support ↑f) (↑f + g) (↑f' + g) x ** simpa only [coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_mem _ _ _ hx, _root_.add_left_inj, add_right_eq_self]
using Set.disjoint_left.1 hff' hx ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 E : Type u_5 x✝ : MeasurableSpace α inst✝² : MeasurableSpace E inst✝¹ : AddGroup E inst✝ : MeasurableAdd E g : α → E hg : Measurable g f f' : α →ₛ E hff' : Disjoint (support ↑f) (support ↑f') hf : Measurable (↑f + g) hf' : Measurable (↑f' + g) x : α hx : ¬x ∈ support ↑f ⊢ (↑(f + f') + g) x = Set.piecewise (support ↑f) (↑f + g) (↑f' + g) x ** simpa only [SimpleFunc.coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_not_mem _ _ _ hx, _root_.add_left_inj, add_left_eq_self] using hx ** Qed
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Measurable.ennreal_induction ** α : Type u_1 inst✝ : MeasurableSpace α P : (α → ℝ≥0∞) → Prop h_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g) h_iSup : ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ n, f n x f : α → ℝ≥0∞ hf : Measurable f ⊢ P f ** convert h_iSup (fun n => (eapprox f n).measurable) (monotone_eapprox f) _ using 1 ** case h.e'_1 α : Type u_1 inst✝ : MeasurableSpace α P : (α → ℝ≥0∞) → Prop h_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g) h_iSup : ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ n, f n x f : α → ℝ≥0∞ hf : Measurable f ⊢ f = fun x => ⨆ n, ↑(eapprox f n) x ** ext1 x ** case h.e'_1.h α : Type u_1 inst✝ : MeasurableSpace α P : (α → ℝ≥0∞) → Prop h_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g) h_iSup : ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ n, f n x f : α → ℝ≥0∞ hf : Measurable f x : α ⊢ f x = ⨆ n, ↑(eapprox f n) x ** rw [iSup_eapprox_apply f hf] ** α : Type u_1 inst✝ : MeasurableSpace α P : (α → ℝ≥0∞) → Prop h_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g) h_iSup : ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ n, f n x f : α → ℝ≥0∞ hf : Measurable f ⊢ ∀ (n : ℕ), P ↑(eapprox f n) ** exact fun n =>
SimpleFunc.induction (fun c s hs => h_ind c hs)
(fun f g hfg hf hg => h_add hfg f.measurable g.measurable hf hg) (eapprox f n) ** Qed
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BoundedContinuousFunction.integrable_of_nnreal ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ : Measure X inst✝ : IsFiniteMeasure μ f : X →ᵇ ℝ≥0 ⊢ Integrable (NNReal.toReal ∘ ↑f) ** refine' ⟨(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable, _⟩ ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ : Measure X inst✝ : IsFiniteMeasure μ f : X →ᵇ ℝ≥0 ⊢ HasFiniteIntegral (NNReal.toReal ∘ ↑f) ** simp only [HasFiniteIntegral, Function.comp_apply, NNReal.nnnorm_eq] ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ : Measure X inst✝ : IsFiniteMeasure μ f : X →ᵇ ℝ≥0 ⊢ ∫⁻ (a : X), ↑(↑f a) ∂μ < ⊤ ** exact lintegral_lt_top_of_nnreal _ f ** Qed
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BoundedContinuousFunction.toReal_lintegral_coe_eq_integral ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ✝ : Measure X inst✝ : IsFiniteMeasure μ✝ f : X →ᵇ ℝ≥0 μ : Measure X ⊢ ENNReal.toReal (∫⁻ (x : X), ↑(↑f x) ∂μ) = ∫ (x : X), ↑(↑f x) ∂μ ** rw [integral_eq_lintegral_of_nonneg_ae _ (by simpa [Function.comp_apply] using
(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable)] ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ✝ : Measure X inst✝ : IsFiniteMeasure μ✝ f : X →ᵇ ℝ≥0 μ : Measure X ⊢ AEStronglyMeasurable (fun x => ↑(↑f x)) μ ** simpa [Function.comp_apply] using
(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ✝ : Measure X inst✝ : IsFiniteMeasure μ✝ f : X →ᵇ ℝ≥0 μ : Measure X ⊢ ENNReal.toReal (∫⁻ (x : X), ↑(↑f x) ∂μ) = ENNReal.toReal (∫⁻ (a : X), ENNReal.ofReal ↑(↑f a) ∂μ) ** simp only [ENNReal.ofReal_coe_nnreal] ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ✝ : Measure X inst✝ : IsFiniteMeasure μ✝ f : X →ᵇ ℝ≥0 μ : Measure X ⊢ 0 ≤ᵐ[μ] fun x => ↑(↑f x) ** exact eventually_of_forall (by simp only [Pi.zero_apply, NNReal.zero_le_coe, imp_true_iff]) ** X : Type u_1 inst✝³ : MeasurableSpace X inst✝² : TopologicalSpace X inst✝¹ : OpensMeasurableSpace X μ✝ : Measure X inst✝ : IsFiniteMeasure μ✝ f : X →ᵇ ℝ≥0 μ : Measure X ⊢ ∀ (x : X), OfNat.ofNat 0 x ≤ (fun x => ↑(↑f x)) x ** simp only [Pi.zero_apply, NNReal.zero_le_coe, imp_true_iff] ** Qed
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ENNReal.fun_eq_funMulInvSnorm_mul_snorm ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ f : α → ℝ≥0∞ hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0 hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ a : α ⊢ f a = funMulInvSnorm f p μ a * (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p) ** simp [funMulInvSnorm, mul_assoc, ENNReal.inv_mul_cancel, hf_nonzero, hf_top] ** Qed
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ENNReal.funMulInvSnorm_rpow ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 < p f : α → ℝ≥0∞ a : α ⊢ funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ** rw [funMulInvSnorm, mul_rpow_of_nonneg _ _ (le_of_lt hp0)] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 < p f : α → ℝ≥0∞ a : α ⊢ f a ^ p * ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ** suffices h_inv_rpow : ((∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ c : α, f c ^ p ∂μ)⁻¹ ** case h_inv_rpow α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 < p f : α → ℝ≥0∞ a : α ⊢ ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ** rw [inv_rpow, ← rpow_mul, one_div_mul_cancel hp0.ne', rpow_one] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 < p f : α → ℝ≥0∞ a : α h_inv_rpow : ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ⊢ f a ^ p * ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ** rw [h_inv_rpow] ** Qed
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ENNReal.ae_eq_zero_of_lintegral_rpow_eq_zero ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 ≤ p f : α → ℝ≥0∞ hf : AEMeasurable f hf_zero : ∫⁻ (a : α), f a ^ p ∂μ = 0 ⊢ f =ᵐ[μ] 0 ** rw [lintegral_eq_zero_iff' (hf.pow_const p)] at hf_zero ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 ≤ p f : α → ℝ≥0∞ hf : AEMeasurable f hf_zero : (fun x => f x ^ p) =ᵐ[μ] 0 ⊢ f =ᵐ[μ] 0 ** refine' Filter.Eventually.mp hf_zero (Filter.eventually_of_forall fun x => _) ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 ≤ p f : α → ℝ≥0∞ hf : AEMeasurable f hf_zero : (fun x => f x ^ p) =ᵐ[μ] 0 x : α ⊢ (fun x => f x ^ p) x = OfNat.ofNat 0 x → f x = OfNat.ofNat 0 x ** dsimp only ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 ≤ p f : α → ℝ≥0∞ hf : AEMeasurable f hf_zero : (fun x => f x ^ p) =ᵐ[μ] 0 x : α ⊢ f x ^ p = OfNat.ofNat 0 x → f x = OfNat.ofNat 0 x ** rw [Pi.zero_apply, ← not_imp_not] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p : ℝ hp0 : 0 ≤ p f : α → ℝ≥0∞ hf : AEMeasurable f hf_zero : (fun x => f x ^ p) =ᵐ[μ] 0 x : α ⊢ ¬f x = 0 → ¬f x ^ p = 0 ** exact fun hx => (rpow_pos_of_nonneg (pos_iff_ne_zero.2 hx) hp0).ne' ** Qed
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ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hp0_lt : 0 < p hq0 : 0 ≤ q f g : α → ℝ≥0∞ hf_top : ∫⁻ (a : α), f a ^ p ∂μ = ⊤ hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0 ⊢ ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q) ** refine' le_trans le_top (le_of_eq _) ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hp0_lt : 0 < p hq0 : 0 ≤ q f g : α → ℝ≥0∞ hf_top : ∫⁻ (a : α), f a ^ p ∂μ = ⊤ hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0 ⊢ ⊤ = (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q) ** have hp0_inv_lt : 0 < 1 / p := by simp [hp0_lt] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hp0_lt : 0 < p hq0 : 0 ≤ q f g : α → ℝ≥0∞ hf_top : ∫⁻ (a : α), f a ^ p ∂μ = ⊤ hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0 hp0_inv_lt : 0 < 1 / p ⊢ ⊤ = (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q) ** rw [hf_top, ENNReal.top_rpow_of_pos hp0_inv_lt] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hp0_lt : 0 < p hq0 : 0 ≤ q f g : α → ℝ≥0∞ hf_top : ∫⁻ (a : α), f a ^ p ∂μ = ⊤ hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0 hp0_inv_lt : 0 < 1 / p ⊢ ⊤ = ⊤ * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q) ** simp [hq0, hg_nonzero] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hp0_lt : 0 < p hq0 : 0 ≤ q f g : α → ℝ≥0∞ hf_top : ∫⁻ (a : α), f a ^ p ∂μ = ⊤ hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0 ⊢ 0 < 1 / p ** simp [hp0_lt] ** Qed
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ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have hp0_ne : p ≠ 0 := (ne_of_lt hp0_lt).symm ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have hp0 : 0 ≤ p := le_of_lt hp0_lt ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have hq0_lt : 0 < q := lt_of_le_of_lt hp0 hpq ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have hq0_ne : q ≠ 0 := (ne_of_lt hq0_lt).symm ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have h_one_div_r : 1 / r = 1 / p - 1 / q := by rw [hpqr]; simp ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have _ : r ≠ 0 := by
have hr_inv_pos : 0 < 1 / r := by rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt]
rw [one_div, _root_.inv_pos] at hr_inv_pos
exact (ne_of_lt hr_inv_pos).symm ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** let p2 := q / p ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** let q2 := p2.conjugateExponent ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have hp2q2 : p2.IsConjugateExponent q2 :=
Real.isConjugateExponent_conjugateExponent (by simp [_root_.lt_div_iff, hpq, hp0_lt]) ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** calc
(∫⁻ a : α, (f * g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ a : α, f a ^ p * g a ^ p ∂μ) ^ (1 / p) := by
simp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0]
_ ≤ ((∫⁻ a, f a ^ (p * p2) ∂μ) ^ (1 / p2) *
(∫⁻ a, g a ^ (p * q2) ∂μ) ^ (1 / q2)) ^ (1 / p) := by
refine' ENNReal.rpow_le_rpow _ (by simp [hp0])
simp_rw [ENNReal.rpow_mul]
exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hp2q2 (hf.pow_const _) (hg.pow_const _)
_ = (∫⁻ a : α, f a ^ q ∂μ) ^ (1 / q) * (∫⁻ a : α, g a ^ r ∂μ) ^ (1 / r) := by
rw [@ENNReal.mul_rpow_of_nonneg _ _ (1 / p) (by simp [hp0]), ← ENNReal.rpow_mul, ←
ENNReal.rpow_mul]
have hpp2 : p * p2 = q := by
symm
rw [mul_comm, ← div_eq_iff hp0_ne]
have hpq2 : p * q2 = r := by
rw [← inv_inv r, ← one_div, ← one_div, h_one_div_r]
field_simp [Real.conjugateExponent, hp0_ne, hq0_ne]
simp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 ⊢ 1 / r = 1 / p - 1 / q ** rw [hpqr] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 ⊢ 1 / r = 1 / q + 1 / r - 1 / q ** simp ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q ⊢ r ≠ 0 ** have hr_inv_pos : 0 < 1 / r := by rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q hr_inv_pos : 0 < 1 / r ⊢ r ≠ 0 ** rw [one_div, _root_.inv_pos] at hr_inv_pos ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q hr_inv_pos : 0 < r ⊢ r ≠ 0 ** exact (ne_of_lt hr_inv_pos).symm ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q ⊢ 0 < 1 / r ** rwa [h_one_div_r, sub_pos, one_div_lt_one_div hq0_lt hp0_lt] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 ⊢ 1 < p2 ** simp [_root_.lt_div_iff, hpq, hp0_lt] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ^ (1 / p) ** simp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ (∫⁻ (a : α), f a ^ p * g a ^ p ∂μ) ^ (1 / p) ≤ ((∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2)) ^ (1 / p) ** refine' ENNReal.rpow_le_rpow _ (by simp [hp0]) ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ ∫⁻ (a : α), f a ^ p * g a ^ p ∂μ ≤ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2) ** simp_rw [ENNReal.rpow_mul] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ ∫⁻ (a : α), f a ^ p * g a ^ p ∂μ ≤ (∫⁻ (a : α), (f a ^ p) ^ (q / p) ∂μ) ^ (1 / (q / p)) * (∫⁻ (a : α), (g a ^ p) ^ Real.conjugateExponent (q / p) ∂μ) ^ (1 / Real.conjugateExponent (q / p)) ** exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hp2q2 (hf.pow_const _) (hg.pow_const _) ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ 0 ≤ 1 / p ** simp [hp0] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ ((∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2)) ^ (1 / p) = (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** rw [@ENNReal.mul_rpow_of_nonneg _ _ (1 / p) (by simp [hp0]), ← ENNReal.rpow_mul, ←
ENNReal.rpow_mul] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) = (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have hpp2 : p * p2 = q := by
symm
rw [mul_comm, ← div_eq_iff hp0_ne] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 hpp2 : p * p2 = q ⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) = (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** have hpq2 : p * q2 = r := by
rw [← inv_inv r, ← one_div, ← one_div, h_one_div_r]
field_simp [Real.conjugateExponent, hp0_ne, hq0_ne] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 hpp2 : p * p2 = q hpq2 : p * q2 = r ⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) = (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) ** simp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ p * p2 = q ** symm ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 ⊢ q = p * p2 ** rw [mul_comm, ← div_eq_iff hp0_ne] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 hpp2 : p * p2 = q ⊢ p * q2 = r ** rw [← inv_inv r, ← one_div, ← one_div, h_one_div_r] ** α✝ : Type u_1 inst✝¹ : MeasurableSpace α✝ μ✝ : Measure α✝ α : Type u_2 inst✝ : MeasurableSpace α p q r : ℝ hp0_lt : 0 < p hpq : p < q hpqr : 1 / p = 1 / q + 1 / r μ : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hp0_ne : p ≠ 0 hp0 : 0 ≤ p hq0_lt : 0 < q hq0_ne : q ≠ 0 h_one_div_r : 1 / r = 1 / p - 1 / q x✝ : r ≠ 0 p2 : ℝ := q / p q2 : ℝ := Real.conjugateExponent p2 hp2q2 : Real.IsConjugateExponent p2 q2 hpp2 : p * p2 = q ⊢ p * q2 = 1 / (1 / p - 1 / q) ** field_simp [Real.conjugateExponent, hp0_ne, hq0_ne] ** Qed
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ENNReal.lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ ⊢ ∫⁻ (a : α), f a * g a ^ (p - 1) ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q) ** refine' le_trans (ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf (hg.pow_const _)) _ ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ ⊢ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (g a ^ (p - 1)) ^ q ∂μ) ^ (1 / q) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q) ** by_cases hf_zero_rpow : (∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) = 0 ** case neg α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 ⊢ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (g a ^ (p - 1)) ^ q ∂μ) ^ (1 / q) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q) ** have hf_top_rpow : (∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
by_contra h
refine' hf_top _
have hp_not_neg : ¬p < 0 := by simp [hpq.nonneg]
simpa [hpq.pos, hp_not_neg] using h ** case neg α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 hf_top_rpow : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) ≠ ⊤ ⊢ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (g a ^ (p - 1)) ^ q ∂μ) ^ (1 / q) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q) ** refine' (ENNReal.mul_le_mul_left hf_zero_rpow hf_top_rpow).mpr (le_of_eq _) ** case neg α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 hf_top_rpow : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) ≠ ⊤ ⊢ (∫⁻ (a : α), (g a ^ (p - 1)) ^ q ∂μ) ^ (1 / q) = (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q) ** congr ** case neg.e_a.e_f α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 hf_top_rpow : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) ≠ ⊤ ⊢ (fun a => (g a ^ (p - 1)) ^ q) = fun a => g a ^ p ** ext1 a ** case neg.e_a.e_f.h α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 hf_top_rpow : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) ≠ ⊤ a : α ⊢ (g a ^ (p - 1)) ^ q = g a ^ p ** rw [← ENNReal.rpow_mul, hpq.sub_one_mul_conj] ** case pos α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 ⊢ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (g a ^ (p - 1)) ^ q ∂μ) ^ (1 / q) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q) ** rw [hf_zero_rpow, zero_mul] ** case pos α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 ⊢ 0 ≤ 0 * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q) ** exact zero_le _ ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 ⊢ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) ≠ ⊤ ** by_contra h ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 h : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = ⊤ ⊢ False ** refine' hf_top _ ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 h : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = ⊤ ⊢ ∫⁻ (a : α), f a ^ p ∂μ = ⊤ ** have hp_not_neg : ¬p < 0 := by simp [hpq.nonneg] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 h : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = ⊤ hp_not_neg : ¬p < 0 ⊢ ∫⁻ (a : α), f a ^ p ∂μ = ⊤ ** simpa [hpq.pos, hp_not_neg] using h ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0 h : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = ⊤ ⊢ ¬p < 0 ** simp [hpq.nonneg] ** Qed
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ENNReal.lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ ⊢ ∫⁻ (a : α), (f + g) a ^ p ∂μ ≤ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ ** refine' lintegral_mono fun a => _ ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α ⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1) ** by_cases h_zero : (f + g) a = 0 ** case neg α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : ¬(f + g) a = 0 ⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1) ** by_cases h_top : (f + g) a = ⊤ ** case neg α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : ¬(f + g) a = 0 h_top : ¬(f + g) a = ⊤ ⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1) ** refine' le_of_eq _ ** case neg α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : ¬(f + g) a = 0 h_top : ¬(f + g) a = ⊤ ⊢ (f + g) a ^ p = (f + g) a * (f + g) a ^ (p - 1) ** nth_rw 2 [← ENNReal.rpow_one ((f + g) a)] ** case neg α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : ¬(f + g) a = 0 h_top : ¬(f + g) a = ⊤ ⊢ (f + g) a ^ p = (f + g) a ^ 1 * (f + g) a ^ (p - 1) ** rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right] ** case pos α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : (f + g) a = 0 ⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1) ** rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos] ** case pos α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : (f + g) a = 0 ⊢ 0 ≤ 0 * 0 ^ (p - 1) ** exact zero_le _ ** case pos α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : ¬(f + g) a = 0 h_top : (f + g) a = ⊤ ⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1) ** rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top] ** case pos α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ a : α h_zero : ¬(f + g) a = 0 h_top : (f + g) a = ⊤ ⊢ ⊤ ^ p ≤ ⊤ ** exact le_top ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ ⊢ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ** have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
(hf.add hg).pow_const _ ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ h_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1) ⊢ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ** have h_add_apply :
(∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
rfl ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ h_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1) h_add_apply : ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), (f a + g a) * (f + g) a ^ (p - 1) ∂μ ⊢ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ** simp_rw [h_add_apply, add_mul] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ h_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1) h_add_apply : ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), (f a + g a) * (f + g) a ^ (p - 1) ∂μ ⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) + g a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ** rw [lintegral_add_left' (hf.mul h_add_m)] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ ⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ≤ ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q) ** rw [add_mul] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0∞ hf : AEMeasurable f hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤ hg : AEMeasurable g hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤ ⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q) ** exact
add_le_add
(lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hf (hf.add hg) hf_top)
(lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hg (hf.add hg) hg_top) ** Qed
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NNReal.lintegral_mul_le_Lp_mul_Lq ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0 hf : AEMeasurable f hg : AEMeasurable g ⊢ ∫⁻ (a : α), ↑((f * g) a) ∂μ ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q) ** simp_rw [Pi.mul_apply, ENNReal.coe_mul] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α p q : ℝ hpq : Real.IsConjugateExponent p q f g : α → ℝ≥0 hf : AEMeasurable f hg : AEMeasurable g ⊢ ∫⁻ (a : α), ↑(f a) * ↑(g a) ∂μ ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q) ** exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf.coe_nnreal_ennreal hg.coe_nnreal_ennreal ** Qed
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IsPiSystem.pi ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 C : (i : ι) → Set (Set (α i)) hC : ∀ (i : ι), IsPiSystem (C i) ⊢ IsPiSystem (Set.pi univ '' Set.pi univ C) ** rintro _ ⟨s₁, hs₁, rfl⟩ _ ⟨s₂, hs₂, rfl⟩ hst ** case intro.intro.intro.intro ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 C : (i : ι) → Set (Set (α i)) hC : ∀ (i : ι), IsPiSystem (C i) s₁ : (i : ι) → Set (α i) hs₁ : s₁ ∈ Set.pi univ C s₂ : (i : ι) → Set (α i) hs₂ : s₂ ∈ Set.pi univ C hst : Set.Nonempty (Set.pi univ s₁ ∩ Set.pi univ s₂) ⊢ Set.pi univ s₁ ∩ Set.pi univ s₂ ∈ Set.pi univ '' Set.pi univ C ** rw [← pi_inter_distrib] at hst ⊢ ** case intro.intro.intro.intro ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 C : (i : ι) → Set (Set (α i)) hC : ∀ (i : ι), IsPiSystem (C i) s₁ : (i : ι) → Set (α i) hs₁ : s₁ ∈ Set.pi univ C s₂ : (i : ι) → Set (α i) hs₂ : s₂ ∈ Set.pi univ C hst : Set.Nonempty (Set.pi univ fun i => s₁ i ∩ s₂ i) ⊢ (Set.pi univ fun i => s₁ i ∩ s₂ i) ∈ Set.pi univ '' Set.pi univ C ** rw [univ_pi_nonempty_iff] at hst ** case intro.intro.intro.intro ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 C : (i : ι) → Set (Set (α i)) hC : ∀ (i : ι), IsPiSystem (C i) s₁ : (i : ι) → Set (α i) hs₁ : s₁ ∈ Set.pi univ C s₂ : (i : ι) → Set (α i) hs₂ : s₂ ∈ Set.pi univ C hst : ∀ (i : ι), Set.Nonempty (s₁ i ∩ s₂ i) ⊢ (Set.pi univ fun i => s₁ i ∩ s₂ i) ∈ Set.pi univ '' Set.pi univ C ** exact mem_image_of_mem _ fun i _ => hC i _ (hs₁ i (mem_univ i)) _ (hs₂ i (mem_univ i)) (hst i) ** Qed
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MeasureTheory.piPremeasure_pi ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) hs : Set.Nonempty (Set.pi univ s) ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** simp [hs, piPremeasure] ** Qed
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MeasureTheory.piPremeasure_pi' ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** cases isEmpty_or_nonempty ι ** case inr ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h✝ : Nonempty ι ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** cases' (pi univ s).eq_empty_or_nonempty with h h ** case inl ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h✝ : IsEmpty ι ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** simp [piPremeasure] ** case inr.inl ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h✝ : Nonempty ι h : Set.pi univ s = ∅ ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** rcases univ_pi_eq_empty_iff.mp h with ⟨i, hi⟩ ** case inr.inl.intro ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h✝ : Nonempty ι h : Set.pi univ s = ∅ i : ι hi : s i = ∅ ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** have : ∃ i, m i (s i) = 0 := ⟨i, by simp [hi]⟩ ** case inr.inl.intro ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h✝ : Nonempty ι h : Set.pi univ s = ∅ i : ι hi : s i = ∅ this : ∃ i, ↑(m i) (s i) = 0 ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** simpa [h, Finset.card_univ, zero_pow (Fintype.card_pos_iff.mpr ‹_›), @eq_comm _ (0 : ℝ≥0∞),
Finset.prod_eq_zero_iff, piPremeasure] ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h✝ : Nonempty ι h : Set.pi univ s = ∅ i : ι hi : s i = ∅ ⊢ ↑(m i) (s i) = 0 ** simp [hi] ** case inr.inr ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h✝ : Nonempty ι h : Set.Nonempty (Set.pi univ s) ⊢ piPremeasure m (Set.pi univ s) = ∏ i : ι, ↑(m i) (s i) ** simp [h, piPremeasure] ** Qed
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MeasureTheory.OuterMeasure.pi_pi_le ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) ⊢ ↑(OuterMeasure.pi m) (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) ** cases' (pi univ s).eq_empty_or_nonempty with h h ** case inl ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h : Set.pi univ s = ∅ ⊢ ↑(OuterMeasure.pi m) (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) case inr ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h : Set.Nonempty (Set.pi univ s) ⊢ ↑(OuterMeasure.pi m) (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) ** simp [h] ** case inr ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) s : (i : ι) → Set (α i) h : Set.Nonempty (Set.pi univ s) ⊢ ↑(OuterMeasure.pi m) (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) ** exact (boundedBy_le _).trans_eq (piPremeasure_pi h) ** Qed
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MeasureTheory.OuterMeasure.le_pi ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) n : OuterMeasure ((i : ι) → α i) ⊢ n ≤ OuterMeasure.pi m ↔ ∀ (s : (i : ι) → Set (α i)), Set.Nonempty (Set.pi univ s) → ↑n (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) ** rw [OuterMeasure.pi, le_boundedBy'] ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) n : OuterMeasure ((i : ι) → α i) ⊢ (∀ (s : Set ((i : ι) → α i)), Set.Nonempty s → ↑n s ≤ piPremeasure m s) ↔ ∀ (s : (i : ι) → Set (α i)), Set.Nonempty (Set.pi univ s) → ↑n (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) ** constructor ** case mp ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) n : OuterMeasure ((i : ι) → α i) ⊢ (∀ (s : Set ((i : ι) → α i)), Set.Nonempty s → ↑n s ≤ piPremeasure m s) → ∀ (s : (i : ι) → Set (α i)), Set.Nonempty (Set.pi univ s) → ↑n (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) ** intro h s hs ** case mp ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) n : OuterMeasure ((i : ι) → α i) h : ∀ (s : Set ((i : ι) → α i)), Set.Nonempty s → ↑n s ≤ piPremeasure m s s : (i : ι) → Set (α i) hs : Set.Nonempty (Set.pi univ s) ⊢ ↑n (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) ** refine' (h _ hs).trans_eq (piPremeasure_pi hs) ** case mpr ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) n : OuterMeasure ((i : ι) → α i) ⊢ (∀ (s : (i : ι) → Set (α i)), Set.Nonempty (Set.pi univ s) → ↑n (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i)) → ∀ (s : Set ((i : ι) → α i)), Set.Nonempty s → ↑n s ≤ piPremeasure m s ** intro h s hs ** case mpr ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) n : OuterMeasure ((i : ι) → α i) h : ∀ (s : (i : ι) → Set (α i)), Set.Nonempty (Set.pi univ s) → ↑n (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) s : Set ((i : ι) → α i) hs : Set.Nonempty s ⊢ ↑n s ≤ piPremeasure m s ** refine' le_trans (n.mono <| subset_pi_eval_image univ s) (h _ _) ** case mpr ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝ : Fintype ι m✝ m : (i : ι) → OuterMeasure (α i) n : OuterMeasure ((i : ι) → α i) h : ∀ (s : (i : ι) → Set (α i)), Set.Nonempty (Set.pi univ s) → ↑n (Set.pi univ s) ≤ ∏ i : ι, ↑(m i) (s i) s : Set ((i : ι) → α i) hs : Set.Nonempty s ⊢ Set.Nonempty (Set.pi univ fun i => eval i '' s) ** simp [univ_pi_nonempty_iff, hs] ** Qed
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MeasureTheory.Measure.pi_univ ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝² : Fintype ι m : (i : ι) → OuterMeasure (α i) inst✝¹ : (i : ι) → MeasurableSpace (α i) μ : (i : ι) → Measure (α i) inst✝ : ∀ (i : ι), SigmaFinite (μ i) ⊢ ↑↑(Measure.pi μ) univ = ∏ i : ι, ↑↑(μ i) univ ** rw [← pi_univ, pi_pi μ] ** Qed
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MeasureTheory.Measure.pi_ball ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝³ : Fintype ι m : (i : ι) → OuterMeasure (α i) inst✝² : (i : ι) → MeasurableSpace (α i) μ : (i : ι) → Measure (α i) inst✝¹ : ∀ (i : ι), SigmaFinite (μ i) inst✝ : (i : ι) → MetricSpace (α i) x : (i : ι) → α i r : ℝ hr : 0 < r ⊢ ↑↑(Measure.pi μ) (Metric.ball x r) = ∏ i : ι, ↑↑(μ i) (Metric.ball (x i) r) ** rw [ball_pi _ hr, pi_pi] ** Qed
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MeasureTheory.Measure.pi_closedBall ** ι : Type u_1 ι' : Type u_2 α : ι → Type u_3 inst✝³ : Fintype ι m : (i : ι) → OuterMeasure (α i) inst✝² : (i : ι) → MeasurableSpace (α i) μ : (i : ι) → Measure (α i) inst✝¹ : ∀ (i : ι), SigmaFinite (μ i) inst✝ : (i : ι) → MetricSpace (α i) x : (i : ι) → α i r : ℝ hr : 0 ≤ r ⊢ ↑↑(Measure.pi μ) (Metric.closedBall x r) = ∏ i : ι, ↑↑(μ i) (Metric.closedBall (x i) r) ** rw [closedBall_pi _ hr, pi_pi] ** Qed
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MeasureTheory.Measure.pi_of_empty ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝ : (i : ι) → OuterMeasure (α✝ i) inst✝² : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝¹ : ∀ (i : ι), SigmaFinite (μ✝ i) α : Type u_4 inst✝ : IsEmpty α β : α → Type u_5 m : (a : α) → MeasurableSpace (β a) μ : (a : α) → Measure (β a) x : optParam ((a : α) → β a) fun a => isEmptyElim a ⊢ Measure.pi μ = dirac x ** haveI : ∀ a, SigmaFinite (μ a) := isEmptyElim ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝ : (i : ι) → OuterMeasure (α✝ i) inst✝² : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝¹ : ∀ (i : ι), SigmaFinite (μ✝ i) α : Type u_4 inst✝ : IsEmpty α β : α → Type u_5 m : (a : α) → MeasurableSpace (β a) μ : (a : α) → Measure (β a) x : optParam ((a : α) → β a) fun a => isEmptyElim a this : ∀ (a : α), SigmaFinite (μ a) ⊢ Measure.pi μ = dirac x ** refine' pi_eq fun s _ => _ ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝ : (i : ι) → OuterMeasure (α✝ i) inst✝² : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝¹ : ∀ (i : ι), SigmaFinite (μ✝ i) α : Type u_4 inst✝ : IsEmpty α β : α → Type u_5 m : (a : α) → MeasurableSpace (β a) μ : (a : α) → Measure (β a) x : optParam ((a : α) → β a) fun a => isEmptyElim a this : ∀ (a : α), SigmaFinite (μ a) s : (i : α) → Set (β i) x✝ : ∀ (i : α), MeasurableSet (s i) ⊢ ↑↑(dirac x) (Set.pi univ s) = ∏ i : α, ↑↑(μ i) (s i) ** rw [Fintype.prod_empty, dirac_apply_of_mem] ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝ : (i : ι) → OuterMeasure (α✝ i) inst✝² : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝¹ : ∀ (i : ι), SigmaFinite (μ✝ i) α : Type u_4 inst✝ : IsEmpty α β : α → Type u_5 m : (a : α) → MeasurableSpace (β a) μ : (a : α) → Measure (β a) x : optParam ((a : α) → β a) fun a => isEmptyElim a this : ∀ (a : α), SigmaFinite (μ a) s : (i : α) → Set (β i) x✝ : ∀ (i : α), MeasurableSet (s i) ⊢ x ∈ Set.pi univ s ** exact isEmptyElim (α := α) ** Qed
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MeasureTheory.measurePreserving_piFinSuccAboveEquiv ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝¹ : (i : ι) → OuterMeasure (α✝ i) m✝ : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝² : ∀ (i : ι), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' n : ℕ α : Fin (n + 1) → Type u m : (i : Fin (n + 1)) → MeasurableSpace (α i) μ : (i : Fin (n + 1)) → Measure (α i) inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i) i : Fin (n + 1) ⊢ MeasurePreserving ↑(MeasurableEquiv.piFinSuccAboveEquiv α i) ** set e := (MeasurableEquiv.piFinSuccAboveEquiv α i).symm ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝¹ : (i : ι) → OuterMeasure (α✝ i) m✝ : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝² : ∀ (i : ι), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' n : ℕ α : Fin (n + 1) → Type u m : (i : Fin (n + 1)) → MeasurableSpace (α i) μ : (i : Fin (n + 1)) → Measure (α i) inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i) i : Fin (n + 1) e : α i × ((j : Fin n) → α (Fin.succAbove i j)) ≃ᵐ ((j : Fin (n + 1)) → α j) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv α i) ⊢ MeasurePreserving ↑(MeasurableEquiv.piFinSuccAboveEquiv α i) ** refine' MeasurePreserving.symm e _ ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝¹ : (i : ι) → OuterMeasure (α✝ i) m✝ : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝² : ∀ (i : ι), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' n : ℕ α : Fin (n + 1) → Type u m : (i : Fin (n + 1)) → MeasurableSpace (α i) μ : (i : Fin (n + 1)) → Measure (α i) inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i) i : Fin (n + 1) e : α i × ((j : Fin n) → α (Fin.succAbove i j)) ≃ᵐ ((j : Fin (n + 1)) → α j) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv α i) ⊢ MeasurePreserving ↑e ** refine' ⟨e.measurable, (pi_eq fun s _ => _).symm⟩ ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝¹ : (i : ι) → OuterMeasure (α✝ i) m✝ : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝² : ∀ (i : ι), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' n : ℕ α : Fin (n + 1) → Type u m : (i : Fin (n + 1)) → MeasurableSpace (α i) μ : (i : Fin (n + 1)) → Measure (α i) inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i) i : Fin (n + 1) e : α i × ((j : Fin n) → α (Fin.succAbove i j)) ≃ᵐ ((j : Fin (n + 1)) → α j) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv α i) s : (i : Fin (n + 1)) → Set (α i) x✝ : ∀ (i : Fin (n + 1)), MeasurableSet (s i) ⊢ ↑↑(Measure.map (↑e) (Measure.prod (μ i) (Measure.pi fun j => μ (Fin.succAbove i j)))) (Set.pi univ s) = ∏ i : Fin (n + 1), ↑↑(μ i) (s i) ** rw [e.map_apply, i.prod_univ_succAbove _, ← pi_pi, ← prod_prod] ** ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝¹ : (i : ι) → OuterMeasure (α✝ i) m✝ : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝² : ∀ (i : ι), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' n : ℕ α : Fin (n + 1) → Type u m : (i : Fin (n + 1)) → MeasurableSpace (α i) μ : (i : Fin (n + 1)) → Measure (α i) inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i) i : Fin (n + 1) e : α i × ((j : Fin n) → α (Fin.succAbove i j)) ≃ᵐ ((j : Fin (n + 1)) → α j) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv α i) s : (i : Fin (n + 1)) → Set (α i) x✝ : ∀ (i : Fin (n + 1)), MeasurableSet (s i) ⊢ ↑↑(Measure.prod (μ i) (Measure.pi fun j => μ (Fin.succAbove i j))) (↑e ⁻¹' Set.pi univ s) = ↑↑(Measure.prod (μ i) (Measure.pi fun i_1 => μ (Fin.succAbove i i_1))) (s i ×ˢ Set.pi univ fun i_1 => s (Fin.succAbove i i_1)) ** congr 1 with ⟨x, f⟩ ** case e_a.h.mk ι : Type u_1 ι' : Type u_2 α✝ : ι → Type u_3 inst✝³ : Fintype ι m✝¹ : (i : ι) → OuterMeasure (α✝ i) m✝ : (i : ι) → MeasurableSpace (α✝ i) μ✝ : (i : ι) → Measure (α✝ i) inst✝² : ∀ (i : ι), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' n : ℕ α : Fin (n + 1) → Type u m : (i : Fin (n + 1)) → MeasurableSpace (α i) μ : (i : Fin (n + 1)) → Measure (α i) inst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i) i : Fin (n + 1) e : α i × ((j : Fin n) → α (Fin.succAbove i j)) ≃ᵐ ((j : Fin (n + 1)) → α j) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv α i) s : (i : Fin (n + 1)) → Set (α i) x✝ : ∀ (i : Fin (n + 1)), MeasurableSet (s i) x : α i f : (j : Fin n) → α (Fin.succAbove i j) ⊢ (x, f) ∈ ↑e ⁻¹' Set.pi univ s ↔ (x, f) ∈ s i ×ˢ Set.pi univ fun i_1 => s (Fin.succAbove i i_1) ** simp [i.forall_iff_succAbove] ** Qed
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MeasureTheory.measurePreserving_pi_empty ** ι✝ : Type u_1 ι' : Type u_2 α✝ : ι✝ → Type u_3 inst✝³ : Fintype ι✝ m✝¹ : (i : ι✝) → OuterMeasure (α✝ i) m✝ : (i : ι✝) → MeasurableSpace (α✝ i) μ✝ : (i : ι✝) → Measure (α✝ i) inst✝² : ∀ (i : ι✝), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' ι : Type u α : ι → Type v inst✝ : IsEmpty ι m : (i : ι) → MeasurableSpace (α i) μ : (i : ι) → Measure (α i) ⊢ MeasurePreserving ↑(MeasurableEquiv.ofUniqueOfUnique ((i : ι) → α i) Unit) ** set e := MeasurableEquiv.ofUniqueOfUnique (∀ i, α i) Unit ** ι✝ : Type u_1 ι' : Type u_2 α✝ : ι✝ → Type u_3 inst✝³ : Fintype ι✝ m✝¹ : (i : ι✝) → OuterMeasure (α✝ i) m✝ : (i : ι✝) → MeasurableSpace (α✝ i) μ✝ : (i : ι✝) → Measure (α✝ i) inst✝² : ∀ (i : ι✝), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' ι : Type u α : ι → Type v inst✝ : IsEmpty ι m : (i : ι) → MeasurableSpace (α i) μ : (i : ι) → Measure (α i) e : ((i : ι) → α i) ≃ᵐ Unit := MeasurableEquiv.ofUniqueOfUnique ((i : ι) → α i) Unit ⊢ MeasurePreserving ↑e ** refine' ⟨e.measurable, _⟩ ** ι✝ : Type u_1 ι' : Type u_2 α✝ : ι✝ → Type u_3 inst✝³ : Fintype ι✝ m✝¹ : (i : ι✝) → OuterMeasure (α✝ i) m✝ : (i : ι✝) → MeasurableSpace (α✝ i) μ✝ : (i : ι✝) → Measure (α✝ i) inst✝² : ∀ (i : ι✝), SigmaFinite (μ✝ i) inst✝¹ : Fintype ι' ι : Type u α : ι → Type v inst✝ : IsEmpty ι m : (i : ι) → MeasurableSpace (α i) μ : (i : ι) → Measure (α i) e : ((i : ι) → α i) ≃ᵐ Unit := MeasurableEquiv.ofUniqueOfUnique ((i : ι) → α i) Unit ⊢ Measure.map (↑e) (Measure.pi μ) = Measure.dirac () ** rw [Measure.pi_of_empty, Measure.map_dirac e.measurable] ** Qed
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MeasureTheory.forall_measure_preimage_mul_iff ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G ⊢ (∀ (g : G), Measure.map (fun x => g * x) μ = μ) ↔ IsMulLeftInvariant μ ** exact ⟨fun h => ⟨h⟩, fun h => h.1⟩ ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G ⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => g * h) ⁻¹' A) = ↑↑μ A) ↔ ∀ (g : G), Measure.map (fun x => g * x) μ = μ ** simp_rw [Measure.ext_iff] ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G ⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => g * h) ⁻¹' A) = ↑↑μ A) ↔ ∀ (g : G) (s : Set G), MeasurableSet s → ↑↑(Measure.map (fun x => g * x) μ) s = ↑↑μ s ** refine' forall_congr' fun g => forall_congr' fun A => forall_congr' fun hA => _ ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G g : G A : Set G hA : MeasurableSet A ⊢ ↑↑μ ((fun h => g * h) ⁻¹' A) = ↑↑μ A ↔ ↑↑(Measure.map (fun x => g * x) μ) A = ↑↑μ A ** rw [map_apply (measurable_const_mul g) hA] ** Qed
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MeasureTheory.forall_measure_preimage_mul_right_iff ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G ⊢ (∀ (g : G), Measure.map (fun x => x * g) μ = μ) ↔ IsMulRightInvariant μ ** exact ⟨fun h => ⟨h⟩, fun h => h.1⟩ ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G ⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔ ∀ (g : G), Measure.map (fun x => x * g) μ = μ ** simp_rw [Measure.ext_iff] ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G ⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔ ∀ (g : G) (s : Set G), MeasurableSet s → ↑↑(Measure.map (fun x => x * g) μ) s = ↑↑μ s ** refine' forall_congr' fun g => forall_congr' fun A => forall_congr' fun hA => _ ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : Mul G μ✝ : Measure G inst✝ : MeasurableMul G μ : Measure G g : G A : Set G hA : MeasurableSet A ⊢ ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A ↔ ↑↑(Measure.map (fun x => x * g) μ) A = ↑↑μ A ** rw [map_apply (measurable_mul_const g) hA] ** Qed
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MeasureTheory.isMulLeftInvariant_map ** 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f ⊢ IsMulLeftInvariant (Measure.map (↑f) μ) ** refine' ⟨fun h => _⟩ ** 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f h : H ⊢ Measure.map (fun x => h * x) (Measure.map (↑f) μ) = Measure.map (↑f) μ ** rw [map_map (measurable_const_mul _) hf] ** 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f h : H ⊢ Measure.map ((fun x => h * x) ∘ ↑f) μ = Measure.map (↑f) μ ** obtain ⟨g, rfl⟩ := h_surj h ** case intro 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f g : G ⊢ Measure.map ((fun x => ↑f g * x) ∘ ↑f) μ = Measure.map (↑f) μ ** conv_rhs => rw [← map_mul_left_eq_self μ g] ** case intro 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f g : G ⊢ Measure.map ((fun x => ↑f g * x) ∘ ↑f) μ = Measure.map (↑f) (Measure.map (fun x => g * x) μ) ** rw [map_map hf (measurable_const_mul _)] ** case intro 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f g : G ⊢ Measure.map ((fun x => ↑f g * x) ∘ ↑f) μ = Measure.map (↑f ∘ fun x => g * x) μ ** congr 2 ** case intro.e_f 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f g : G ⊢ (fun x => ↑f g * x) ∘ ↑f = ↑f ∘ fun x => g * x ** ext y ** case intro.e_f.h 𝕜 : Type u_1 G : Type u_2 H✝ : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H✝ inst✝⁵ : Mul G μ : Measure G inst✝⁴ : MeasurableMul G H : Type u_4 inst✝³ : MeasurableSpace H inst✝² : Mul H inst✝¹ : MeasurableMul H inst✝ : IsMulLeftInvariant μ f : G →ₙ* H hf : Measurable ↑f h_surj : Surjective ↑f g y : G ⊢ ((fun x => ↑f g * x) ∘ ↑f) y = (↑f ∘ fun x => g * x) y ** simp only [comp_apply, map_mul] ** Qed
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MeasureTheory.map_div_right_eq_self ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝³ : MeasurableSpace G inst✝² : MeasurableSpace H inst✝¹ : DivInvMonoid G μ : Measure G inst✝ : IsMulRightInvariant μ g : G ⊢ Measure.map (fun x => x / g) μ = μ ** simp_rw [div_eq_mul_inv, map_mul_right_eq_self μ g⁻¹] ** Qed
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MeasureTheory.measurePreserving_div_right ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableSpace H inst✝² : Group G inst✝¹ : MeasurableMul G μ : Measure G inst✝ : IsMulRightInvariant μ g : G ⊢ MeasurePreserving fun x => x / g ** simp_rw [div_eq_mul_inv, measurePreserving_mul_right μ g⁻¹] ** Qed
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MeasureTheory.measure_preimage_mul ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableSpace H inst✝² : Group G inst✝¹ : MeasurableMul G μ : Measure G inst✝ : IsMulLeftInvariant μ g : G A : Set G ⊢ ↑↑(Measure.map (fun h => g * h) μ) A = ↑↑μ A ** rw [map_mul_left_eq_self μ g] ** Qed
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MeasureTheory.measure_preimage_mul_right ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁴ : MeasurableSpace G inst✝³ : MeasurableSpace H inst✝² : Group G inst✝¹ : MeasurableMul G μ : Measure G inst✝ : IsMulRightInvariant μ g : G A : Set G ⊢ ↑↑(Measure.map (fun h => h * g) μ) A = ↑↑μ A ** rw [map_mul_right_eq_self μ g] ** Qed
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MeasureTheory.Measure.measurePreserving_div_left ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : DivisionMonoid G inst✝³ : MeasurableMul G inst✝² : MeasurableInv G μ✝ μ : Measure G inst✝¹ : IsInvInvariant μ inst✝ : IsMulLeftInvariant μ g : G ⊢ MeasurePreserving fun t => g / t ** simp_rw [div_eq_mul_inv] ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : DivisionMonoid G inst✝³ : MeasurableMul G inst✝² : MeasurableInv G μ✝ μ : Measure G inst✝¹ : IsInvInvariant μ inst✝ : IsMulLeftInvariant μ g : G ⊢ MeasurePreserving fun t => g * t⁻¹ ** exact (measurePreserving_mul_left μ g).comp (measurePreserving_inv μ) ** Qed
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MeasureTheory.regular_inv_iff ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : T2Space G ⊢ Regular (Measure.inv μ) ↔ Regular μ ** constructor ** case mp 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : T2Space G ⊢ Regular (Measure.inv μ) → Regular μ ** intro h ** case mp 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : T2Space G h : Regular (Measure.inv μ) ⊢ Regular μ ** rw [← μ.inv_inv] ** case mp 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : T2Space G h : Regular (Measure.inv μ) ⊢ Regular (Measure.inv (Measure.inv μ)) ** exact Measure.Regular.inv ** case mpr 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : T2Space G ⊢ Regular μ → Regular (Measure.inv μ) ** intro h ** case mpr 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : T2Space G h : Regular μ ⊢ Regular (Measure.inv μ) ** exact Measure.Regular.inv ** Qed
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MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_compact ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ K : Set G hK : IsCompact K h : ↑↑μ K ≠ 0 ⊢ IsOpenPosMeasure μ ** refine' ⟨fun U hU hne => _⟩ ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ K : Set G hK : IsCompact K h : ↑↑μ K ≠ 0 U : Set G hU : IsOpen U hne : Set.Nonempty U ⊢ ↑↑μ U ≠ 0 ** contrapose! h ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ K : Set G hK : IsCompact K U : Set G hU : IsOpen U hne : Set.Nonempty U h : ↑↑μ U = 0 ⊢ ↑↑μ K = 0 ** rw [← nonpos_iff_eq_zero] ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ K : Set G hK : IsCompact K U : Set G hU : IsOpen U hne : Set.Nonempty U h : ↑↑μ U = 0 ⊢ ↑↑μ K ≤ 0 ** rw [← hU.interior_eq] at hne ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ K : Set G hK : IsCompact K U : Set G hU : IsOpen U hne : Set.Nonempty (interior U) h : ↑↑μ U = 0 ⊢ ↑↑μ K ≤ 0 ** obtain ⟨t, hKt⟩ : ∃ t : Finset G, K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U :=
compact_covered_by_mul_left_translates hK hne ** case intro 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ K : Set G hK : IsCompact K U : Set G hU : IsOpen U hne : Set.Nonempty (interior U) h : ↑↑μ U = 0 t : Finset G hKt : K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' U ⊢ ↑↑μ K ≤ 0 ** calc
μ K ≤ μ (⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U) := measure_mono hKt
_ ≤ ∑ g in t, μ ((fun h : G => g * h) ⁻¹' U) := (measure_biUnion_finset_le _ _)
_ = 0 := by simp [measure_preimage_mul, h] ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ K : Set G hK : IsCompact K U : Set G hU : IsOpen U hne : Set.Nonempty (interior U) h : ↑↑μ U = 0 t : Finset G hKt : K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' U ⊢ ∑ g in t, ↑↑μ ((fun h => g * h) ⁻¹' U) = 0 ** simp [measure_preimage_mul, h] ** Qed
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MeasureTheory.null_iff_of_isMulLeftInvariant ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H inst✝⁵ : TopologicalSpace G inst✝⁴ : BorelSpace G μ : Measure G inst✝³ : Group G inst✝² : TopologicalGroup G inst✝¹ : IsMulLeftInvariant μ inst✝ : Regular μ s : Set G hs : IsOpen s ⊢ ↑↑μ s = 0 ↔ s = ∅ ∨ μ = 0 ** by_cases h3μ : μ = 0 ** case pos 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H inst✝⁵ : TopologicalSpace G inst✝⁴ : BorelSpace G μ : Measure G inst✝³ : Group G inst✝² : TopologicalGroup G inst✝¹ : IsMulLeftInvariant μ inst✝ : Regular μ s : Set G hs : IsOpen s h3μ : μ = 0 ⊢ ↑↑μ s = 0 ↔ s = ∅ ∨ μ = 0 ** simp [h3μ] ** case neg 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H inst✝⁵ : TopologicalSpace G inst✝⁴ : BorelSpace G μ : Measure G inst✝³ : Group G inst✝² : TopologicalGroup G inst✝¹ : IsMulLeftInvariant μ inst✝ : Regular μ s : Set G hs : IsOpen s h3μ : ¬μ = 0 ⊢ ↑↑μ s = 0 ↔ s = ∅ ∨ μ = 0 ** haveI := isOpenPosMeasure_of_mulLeftInvariant_of_regular h3μ ** case neg 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁷ : MeasurableSpace G inst✝⁶ : MeasurableSpace H inst✝⁵ : TopologicalSpace G inst✝⁴ : BorelSpace G μ : Measure G inst✝³ : Group G inst✝² : TopologicalGroup G inst✝¹ : IsMulLeftInvariant μ inst✝ : Regular μ s : Set G hs : IsOpen s h3μ : ¬μ = 0 this : IsOpenPosMeasure μ ⊢ ↑↑μ s = 0 ↔ s = ∅ ∨ μ = 0 ** simp only [h3μ, or_false_iff, hs.measure_eq_zero_iff μ] ** Qed
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MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ U : Set G hU : IsOpen U h'U : Set.Nonempty U h : ↑↑μ U ≠ ⊤ K : Set G hK : IsCompact K ⊢ ↑↑μ K < ⊤ ** rw [← hU.interior_eq] at h'U ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ U : Set G hU : IsOpen U h'U : Set.Nonempty (interior U) h : ↑↑μ U ≠ ⊤ K : Set G hK : IsCompact K ⊢ ↑↑μ K < ⊤ ** obtain ⟨t, hKt⟩ : ∃ t : Finset G, K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U :=
compact_covered_by_mul_left_translates hK h'U ** case intro 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ U : Set G hU : IsOpen U h'U : Set.Nonempty (interior U) h : ↑↑μ U ≠ ⊤ K : Set G hK : IsCompact K t : Finset G hKt : K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' U ⊢ ↑↑μ K < ⊤ ** calc
μ K ≤ μ (⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U) := measure_mono hKt
_ ≤ ∑ g in t, μ ((fun h : G => g * h) ⁻¹' U) := (measure_biUnion_finset_le _ _)
_ = Finset.card t * μ U := by simp only [measure_preimage_mul, Finset.sum_const, nsmul_eq_mul]
_ < ∞ := ENNReal.mul_lt_top (ENNReal.nat_ne_top _) h ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : TopologicalSpace G inst✝³ : BorelSpace G μ : Measure G inst✝² : Group G inst✝¹ : TopologicalGroup G inst✝ : IsMulLeftInvariant μ U : Set G hU : IsOpen U h'U : Set.Nonempty (interior U) h : ↑↑μ U ≠ ⊤ K : Set G hK : IsCompact K t : Finset G hKt : K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' U ⊢ ∑ g in t, ↑↑μ ((fun h => g * h) ⁻¹' U) = ↑(Finset.card t) * ↑↑μ U ** simp only [measure_preimage_mul, Finset.sum_const, nsmul_eq_mul] ** Qed
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MeasureTheory.Measure.haar_singleton ** 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : Group G inst✝³ : TopologicalSpace G μ : Measure G inst✝² : IsHaarMeasure μ inst✝¹ : TopologicalGroup G inst✝ : BorelSpace G g : G ⊢ ↑↑μ {g} = ↑↑μ {1} ** convert measure_preimage_mul μ g⁻¹ _ ** case h.e'_2.h.e'_3 𝕜 : Type u_1 G : Type u_2 H : Type u_3 inst✝⁶ : MeasurableSpace G inst✝⁵ : MeasurableSpace H inst✝⁴ : Group G inst✝³ : TopologicalSpace G μ : Measure G inst✝² : IsHaarMeasure μ inst✝¹ : TopologicalGroup G inst✝ : BorelSpace G g : G ⊢ {g} = (fun h => g⁻¹ * h) ⁻¹' {1} ** simp only [mul_one, preimage_mul_left_singleton, inv_inv] ** Qed
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MeasureTheory.laverage_zero ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ ⊢ ⨍⁻ (_x : α), 0 ∂μ = 0 ** rw [laverage, lintegral_zero] ** Qed
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MeasureTheory.laverage_zero_measure ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g f : α → ℝ≥0∞ ⊢ ⨍⁻ (x : α), f x ∂0 = 0 ** simp [laverage] ** Qed
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MeasureTheory.laverage_eq ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g f : α → ℝ≥0∞ ⊢ ⨍⁻ (x : α), f x ∂μ = (∫⁻ (x : α), f x ∂μ) / ↑↑μ univ ** rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul] ** Qed
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MeasureTheory.laverage_eq_lintegral ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g : α → ℝ≥0∞ inst✝ : IsProbabilityMeasure μ f : α → ℝ≥0∞ ⊢ ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ ** rw [laverage, measure_univ, inv_one, one_smul] ** Qed
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MeasureTheory.setLaverage_eq ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t : Set α f✝ g f : α → ℝ≥0∞ s : Set α ⊢ ⨍⁻ (x : α) in s, f x ∂μ = (∫⁻ (x : α) in s, f x ∂μ) / ↑↑μ s ** rw [laverage_eq, restrict_apply_univ] ** Qed
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MeasureTheory.setLaverage_eq' ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t : Set α f✝ g f : α → ℝ≥0∞ s : Set α ⊢ ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α), f x ∂(↑↑μ s)⁻¹ • Measure.restrict μ s ** simp only [laverage_eq', restrict_apply_univ] ** Qed
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MeasureTheory.laverage_congr ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g✝ f g : α → ℝ≥0∞ h : f =ᶠ[ae μ] g ⊢ ⨍⁻ (x : α), f x ∂μ = ⨍⁻ (x : α), g x ∂μ ** simp only [laverage_eq, lintegral_congr_ae h] ** Qed
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MeasureTheory.setLaverage_congr_fun ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hs : MeasurableSet s h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x ⊢ ⨍⁻ (x : α) in s, f x ∂μ = ⨍⁻ (x : α) in s, g x ∂μ ** simp only [laverage_eq, set_lintegral_congr_fun hs h] ** Qed
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MeasureTheory.laverage_lt_top ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ ⊢ ⨍⁻ (x : α), f x ∂μ < ⊤ ** obtain rfl | hμ := eq_or_ne μ 0 ** case inl α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F ν : Measure α s t : Set α f g : α → ℝ≥0∞ hf : ∫⁻ (x : α), f x ∂0 ≠ ⊤ ⊢ ⨍⁻ (x : α), f x ∂0 < ⊤ ** simp ** case inr α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hμ : μ ≠ 0 ⊢ ⨍⁻ (x : α), f x ∂μ < ⊤ ** rw [laverage_eq] ** case inr α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hμ : μ ≠ 0 ⊢ (∫⁻ (x : α), f x ∂μ) / ↑↑μ univ < ⊤ ** exact div_lt_top hf (measure_univ_ne_zero.2 hμ) ** Qed
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MeasureTheory.measure_mul_setLaverage ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g f : α → ℝ≥0∞ h : ↑↑μ s ≠ ⊤ ⊢ ↑↑μ s * ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, f x ∂μ ** have := Fact.mk h.lt_top ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g f : α → ℝ≥0∞ h : ↑↑μ s ≠ ⊤ this : Fact (↑↑μ s < ⊤) ⊢ ↑↑μ s * ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, f x ∂μ ** rw [←measure_mul_laverage, restrict_apply_univ] ** Qed
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MeasureTheory.laverage_union_mem_openSegment ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hs₀ : ↑↑μ s ≠ 0 ht₀ : ↑↑μ t ≠ 0 hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ ⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in t, f x ∂μ) ** refine'
⟨μ s / (μ s + μ t), μ t / (μ s + μ t), ENNReal.div_pos hs₀ <| add_ne_top.2 ⟨hsμ, htμ⟩,
ENNReal.div_pos ht₀ <| add_ne_top.2 ⟨hsμ, htμ⟩, _, (laverage_union hd ht).symm⟩ ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hs₀ : ↑↑μ s ≠ 0 ht₀ : ↑↑μ t ≠ 0 hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ ⊢ ↑↑μ s / (↑↑μ s + ↑↑μ t) + ↑↑μ t / (↑↑μ s + ↑↑μ t) = 1 ** rw [←ENNReal.add_div,
ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)] ** Qed
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MeasureTheory.laverage_union_mem_segment ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ ⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ-[ℝ≥0∞]⨍⁻ (x : α) in t, f x ∂μ] ** by_cases hs₀ : μ s = 0 ** case pos α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hs₀ : ↑↑μ s = 0 ⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ-[ℝ≥0∞]⨍⁻ (x : α) in t, f x ∂μ] ** rw [←ae_eq_empty] at hs₀ ** case pos α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hs₀ : s =ᶠ[ae μ] ∅ ⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ-[ℝ≥0∞]⨍⁻ (x : α) in t, f x ∂μ] ** rw [restrict_congr_set (hs₀.union EventuallyEq.rfl), empty_union] ** case pos α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hs₀ : s =ᶠ[ae μ] ∅ ⊢ ⨍⁻ (x : α) in t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ-[ℝ≥0∞]⨍⁻ (x : α) in t, f x ∂μ] ** exact right_mem_segment _ _ _ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hs₀ : ¬↑↑μ s = 0 ⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ-[ℝ≥0∞]⨍⁻ (x : α) in t, f x ∂μ] ** refine'
⟨μ s / (μ s + μ t), μ t / (μ s + μ t), zero_le _, zero_le _, _, (laverage_union hd ht).symm⟩ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hs₀ : ¬↑↑μ s = 0 ⊢ ↑↑μ s / (↑↑μ s + ↑↑μ t) + ↑↑μ t / (↑↑μ s + ↑↑μ t) = 1 ** rw [←ENNReal.add_div,
ENNReal.div_self (add_eq_zero.not.2 fun h => hs₀ h.1) (add_ne_top.2 ⟨hsμ, htμ⟩)] ** Qed
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MeasureTheory.laverage_const ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f g : α → ℝ≥0∞ μ : Measure α inst✝ : IsFiniteMeasure μ h : NeZero μ c : ℝ≥0∞ ⊢ ⨍⁻ (_x : α), c ∂μ = c ** simp only [laverage, lintegral_const, measure_univ, mul_one] ** Qed
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MeasureTheory.setLaverage_const ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → ℝ≥0∞ hs₀ : ↑↑μ s ≠ 0 hs : ↑↑μ s ≠ ⊤ c : ℝ≥0∞ ⊢ ⨍⁻ (_x : α) in s, c ∂μ = c ** simp only [setLaverage_eq, lintegral_const, Measure.restrict_apply, MeasurableSet.univ,
univ_inter, div_eq_mul_inv, mul_assoc, ENNReal.mul_inv_cancel hs₀ hs, mul_one] ** Qed
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MeasureTheory.lintegral_laverage ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → ℝ≥0∞ μ : Measure α inst✝ : IsFiniteMeasure μ f : α → ℝ≥0∞ ⊢ ∫⁻ (_x : α), ⨍⁻ (a : α), f a ∂μ ∂μ = ∫⁻ (x : α), f x ∂μ ** obtain rfl | hμ := eq_or_ne μ 0 ** case inl α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g f : α → ℝ≥0∞ inst✝ : IsFiniteMeasure 0 ⊢ ∫⁻ (_x : α), ⨍⁻ (a : α), f a ∂0 ∂0 = ∫⁻ (x : α), f x ∂0 ** simp ** case inr α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → ℝ≥0∞ μ : Measure α inst✝ : IsFiniteMeasure μ f : α → ℝ≥0∞ hμ : μ ≠ 0 ⊢ ∫⁻ (_x : α), ⨍⁻ (a : α), f a ∂μ ∂μ = ∫⁻ (x : α), f x ∂μ ** rw [lintegral_const, laverage_eq,
ENNReal.div_mul_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)] ** Qed
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MeasureTheory.average_zero ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → E ⊢ ⨍ (x : α), 0 ∂μ = 0 ** rw [average, integral_zero] ** Qed
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MeasureTheory.average_zero_measure ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g f : α → E ⊢ ⨍ (x : α), f x ∂0 = 0 ** rw [average, smul_zero, integral_zero_measure] ** Qed
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MeasureTheory.average_eq ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g f : α → E ⊢ ⨍ (x : α), f x ∂μ = (ENNReal.toReal (↑↑μ univ))⁻¹ • ∫ (x : α), f x ∂μ ** rw [average_eq', integral_smul_measure, ENNReal.toReal_inv] ** Qed
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MeasureTheory.average_eq_integral ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g : α → E inst✝ : IsProbabilityMeasure μ f : α → E ⊢ ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ ** rw [average, measure_univ, inv_one, one_smul] ** Qed
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MeasureTheory.setAverage_eq ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t : Set α f✝ g f : α → E s : Set α ⊢ ⨍ (x : α) in s, f x ∂μ = (ENNReal.toReal (↑↑μ s))⁻¹ • ∫ (x : α) in s, f x ∂μ ** rw [average_eq, restrict_apply_univ] ** Qed
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MeasureTheory.setAverage_eq' ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t : Set α f✝ g f : α → E s : Set α ⊢ ⨍ (x : α) in s, f x ∂μ = ∫ (x : α), f x ∂(↑↑μ s)⁻¹ • Measure.restrict μ s ** simp only [average_eq', restrict_apply_univ] ** Qed
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MeasureTheory.average_congr ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g ⊢ ⨍ (x : α), f x ∂μ = ⨍ (x : α), g x ∂μ ** simp only [average_eq, integral_congr_ae h] ** Qed
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MeasureTheory.setAverage_congr ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → E h : s =ᶠ[ae μ] t ⊢ ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in t, f x ∂μ ** simp only [setAverage_eq, set_integral_congr_set_ae h, measure_congr h] ** Qed
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MeasureTheory.setAverage_congr_fun ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → E hs : MeasurableSet s h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x ⊢ ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in s, g x ∂μ ** simp only [average_eq, set_integral_congr_ae hs h] ** Qed
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MeasureTheory.average_add_measure ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F inst✝² : CompleteSpace F μ ν✝ : Measure α s t : Set α f✝ g : α → E inst✝¹ : IsFiniteMeasure μ ν : Measure α inst✝ : IsFiniteMeasure ν f : α → E hμ : Integrable f hν : Integrable f ⊢ ⨍ (x : α), f x ∂(μ + ν) = (ENNReal.toReal (↑↑μ univ) / (ENNReal.toReal (↑↑μ univ) + ENNReal.toReal (↑↑ν univ))) • ⨍ (x : α), f x ∂μ + (ENNReal.toReal (↑↑ν univ) / (ENNReal.toReal (↑↑μ univ) + ENNReal.toReal (↑↑ν univ))) • ⨍ (x : α), f x ∂ν ** simp only [div_eq_inv_mul, mul_smul, measure_smul_average, ←smul_add,
←integral_add_measure hμ hν, ←ENNReal.toReal_add (measure_ne_top μ _) (measure_ne_top ν _)] ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F inst✝² : CompleteSpace F μ ν✝ : Measure α s t : Set α f✝ g : α → E inst✝¹ : IsFiniteMeasure μ ν : Measure α inst✝ : IsFiniteMeasure ν f : α → E hμ : Integrable f hν : Integrable f ⊢ ⨍ (x : α), f x ∂(μ + ν) = (ENNReal.toReal (↑↑μ univ + ↑↑ν univ))⁻¹ • ∫ (x : α), f x ∂(μ + ν) ** rw [average_eq, Measure.add_apply] ** Qed
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MeasureTheory.measure_smul_setAverage ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t : Set α f✝ g f : α → E s : Set α h : ↑↑μ s ≠ ⊤ ⊢ ENNReal.toReal (↑↑μ s) • ⨍ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ ** haveI := Fact.mk h.lt_top ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t : Set α f✝ g f : α → E s : Set α h : ↑↑μ s ≠ ⊤ this : Fact (↑↑μ s < ⊤) ⊢ ENNReal.toReal (↑↑μ s) • ⨍ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ ** rw [←measure_smul_average, restrict_apply_univ] ** Qed
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MeasureTheory.average_union_mem_openSegment ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hs₀ : ↑↑μ s ≠ 0 ht₀ : ↑↑μ t ≠ 0 hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ) ** replace hs₀ : 0 < (μ s).toReal ** case hs₀ α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hs₀ : ↑↑μ s ≠ 0 ht₀ : ↑↑μ t ≠ 0 hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t ⊢ 0 < ENNReal.toReal (↑↑μ s) α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t ht₀ : ↑↑μ t ≠ 0 hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hs₀ : 0 < ENNReal.toReal (↑↑μ s) ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ) ** exact ENNReal.toReal_pos hs₀ hsμ ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t ht₀ : ↑↑μ t ≠ 0 hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hs₀ : 0 < ENNReal.toReal (↑↑μ s) ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ) ** replace ht₀ : 0 < (μ t).toReal ** case ht₀ α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t ht₀ : ↑↑μ t ≠ 0 hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hs₀ : 0 < ENNReal.toReal (↑↑μ s) ⊢ 0 < ENNReal.toReal (↑↑μ t) α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hs₀ : 0 < ENNReal.toReal (↑↑μ s) ht₀ : 0 < ENNReal.toReal (↑↑μ t) ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ) ** exact ENNReal.toReal_pos ht₀ htμ ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hs₀ : 0 < ENNReal.toReal (↑↑μ s) ht₀ : 0 < ENNReal.toReal (↑↑μ t) ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ) ** refine' mem_openSegment_iff_div.mpr
⟨(μ s).toReal, (μ t).toReal, hs₀, ht₀, (average_union hd ht hsμ htμ hfs hft).symm⟩ ** Qed
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MeasureTheory.average_union_mem_segment ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ-[ℝ]⨍ (x : α) in t, f x ∂μ] ** by_cases hse : μ s = 0 ** case pos α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hse : ↑↑μ s = 0 ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ-[ℝ]⨍ (x : α) in t, f x ∂μ] ** rw [←ae_eq_empty] at hse ** case pos α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hse : s =ᶠ[ae μ] ∅ ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ-[ℝ]⨍ (x : α) in t, f x ∂μ] ** rw [restrict_congr_set (hse.union EventuallyEq.rfl), empty_union] ** case pos α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hse : s =ᶠ[ae μ] ∅ ⊢ ⨍ (x : α) in t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ-[ℝ]⨍ (x : α) in t, f x ∂μ] ** exact right_mem_segment _ _ _ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hse : ¬↑↑μ s = 0 ⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ-[ℝ]⨍ (x : α) in t, f x ∂μ] ** refine'
mem_segment_iff_div.mpr
⟨(μ s).toReal, (μ t).toReal, ENNReal.toReal_nonneg, ENNReal.toReal_nonneg, _,
(average_union hd ht hsμ htμ hfs hft).symm⟩ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F μ ν : Measure α s✝ t✝ : Set α f✝ g f : α → E s t : Set α hd : AEDisjoint μ s t ht : NullMeasurableSet t hsμ : ↑↑μ s ≠ ⊤ htμ : ↑↑μ t ≠ ⊤ hfs : IntegrableOn f s hft : IntegrableOn f t hse : ¬↑↑μ s = 0 ⊢ 0 < ENNReal.toReal (↑↑μ s) + ENNReal.toReal (↑↑μ t) ** calc
0 < (μ s).toReal := ENNReal.toReal_pos hse hsμ
_ ≤ _ := le_add_of_nonneg_right ENNReal.toReal_nonneg ** Qed
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MeasureTheory.average_const ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f g : α → E μ : Measure α inst✝ : IsFiniteMeasure μ h : NeZero μ c : E ⊢ ⨍ (_x : α), c ∂μ = c ** rw [average, integral_const, measure_univ, ENNReal.one_toReal, one_smul] ** Qed
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MeasureTheory.integral_average ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → E μ : Measure α inst✝ : IsFiniteMeasure μ f : α → E ⊢ ∫ (x : α), ⨍ (a : α), f a ∂μ ∂μ = ∫ (x : α), f x ∂μ ** simp ** Qed
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MeasureTheory.integral_sub_average ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → E μ : Measure α inst✝ : IsFiniteMeasure μ f : α → E ⊢ ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0 ** by_cases hf : Integrable f μ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → E μ : Measure α inst✝ : IsFiniteMeasure μ f : α → E hf : ¬Integrable f ⊢ ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0 ** refine integral_undef fun h => hf ?_ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → E μ : Measure α inst✝ : IsFiniteMeasure μ f : α → E hf : ¬Integrable f h : Integrable fun x => f x - ⨍ (a : α), f a ∂μ ⊢ Integrable f ** convert h.add (integrable_const (⨍ a, f a ∂μ)) ** case h.e'_5 α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → E μ : Measure α inst✝ : IsFiniteMeasure μ f : α → E hf : ¬Integrable f h : Integrable fun x => f x - ⨍ (a : α), f a ∂μ ⊢ f = (fun x => f x - ⨍ (a : α), f a ∂μ) + fun x => ⨍ (a : α), f a ∂μ ** exact (sub_add_cancel _ _).symm ** case pos α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ✝ ν : Measure α s t : Set α f✝ g : α → E μ : Measure α inst✝ : IsFiniteMeasure μ f : α → E hf : Integrable f ⊢ ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0 ** rw [integral_sub hf (integrable_const _), integral_average, sub_self] ** Qed
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MeasureTheory.integral_average_sub ** α : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace α inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E inst✝³ : NormedAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F μ ν : Measure α s t : Set α f g : α → E inst✝ : IsFiniteMeasure μ hf : Integrable f ⊢ ∫ (x : α), ⨍ (a : α), f a ∂μ - f x ∂μ = 0 ** rw [integral_sub (integrable_const _) hf, integral_average, sub_self] ** Qed
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