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Dense.borel_eq_generateFrom_Ioc_mem_aux ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²⁰ : TopologicalSpace α✝ inst✝¹⁹ : MeasurableSpace α✝ inst✝¹⁸ : OpensMeasurableSpace α✝ inst✝¹⁷ : TopologicalSpace β inst✝¹⁶ : MeasurableSpace β inst✝¹⁵ : OpensMeasurableSpace β inst✝¹⁴ : TopologicalSpace γ inst✝¹³ : MeasurableSpace γ inst✝¹² : BorelSpace γ inst✝¹¹ : TopologicalSpace γ₂ inst✝¹⁰ : MeasurableSpace γ₂ inst✝⁹ : BorelSpace γ₂ inst✝⁸ : MeasurableSpace δ α' : Type u_6 inst✝⁷ : TopologicalSpace α' inst✝⁶ : MeasurableSpace α' inst✝⁵ : LinearOrder α✝ inst✝⁴ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α hd : Dense s hbot : ∀ (x : α), IsTop x → x ∈ s hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s ⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ioc l u = S} ** convert hd.orderDual.borel_eq_generateFrom_Ico_mem_aux hbot fun x y hlt he => hIoo y x hlt _
using 2 ** case h.e'_3.h.h.e'_2.h α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²⁰ : TopologicalSpace α✝ inst✝¹⁹ : MeasurableSpace α✝ inst✝¹⁸ : OpensMeasurableSpace α✝ inst✝¹⁷ : TopologicalSpace β inst✝¹⁶ : MeasurableSpace β inst✝¹⁵ : OpensMeasurableSpace β inst✝¹⁴ : TopologicalSpace γ inst✝¹³ : MeasurableSpace γ inst✝¹² : BorelSpace γ inst✝¹¹ : TopologicalSpace γ₂ inst✝¹⁰ : MeasurableSpace γ₂ inst✝⁹ : BorelSpace γ₂ inst✝⁸ : MeasurableSpace δ α' : Type u_6 inst✝⁷ : TopologicalSpace α' inst✝⁶ : MeasurableSpace α' inst✝⁵ : LinearOrder α✝ inst✝⁴ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α hd : Dense s hbot : ∀ (x : α), IsTop x → x ∈ s hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ e_1✝ : α = αᵒᵈ ⊢ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ioc l u = S} = {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s ∧ l < u ∧ Ico l u = S} ** ext s ** case h.e'_3.h.h.e'_2.h.h α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝¹ t u : Set α✝ inst✝²⁰ : TopologicalSpace α✝ inst✝¹⁹ : MeasurableSpace α✝ inst✝¹⁸ : OpensMeasurableSpace α✝ inst✝¹⁷ : TopologicalSpace β inst✝¹⁶ : MeasurableSpace β inst✝¹⁵ : OpensMeasurableSpace β inst✝¹⁴ : TopologicalSpace γ inst✝¹³ : MeasurableSpace γ inst✝¹² : BorelSpace γ inst✝¹¹ : TopologicalSpace γ₂ inst✝¹⁰ : MeasurableSpace γ₂ inst✝⁹ : BorelSpace γ₂ inst✝⁸ : MeasurableSpace δ α' : Type u_6 inst✝⁷ : TopologicalSpace α' inst✝⁶ : MeasurableSpace α' inst✝⁵ : LinearOrder α✝ inst✝⁴ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s✝ : Set α hd : Dense s✝ hbot : ∀ (x : α), IsTop x → x ∈ s✝ hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s✝ e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ e_1✝ : α = αᵒᵈ s : Set α ⊢ s ∈ {S | ∃ l, l ∈ s✝ ∧ ∃ u, u ∈ s✝ ∧ l < u ∧ Ioc l u = S} ↔ s ∈ {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s✝ ∧ l < u ∧ Ico l u = S} ** constructor <;> rintro ⟨l, hl, u, hu, hlt, rfl⟩ ** case h.e'_3.h.h.e'_2.h.h.mp.intro.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α✝ inst✝²⁰ : TopologicalSpace α✝ inst✝¹⁹ : MeasurableSpace α✝ inst✝¹⁸ : OpensMeasurableSpace α✝ inst✝¹⁷ : TopologicalSpace β inst✝¹⁶ : MeasurableSpace β inst✝¹⁵ : OpensMeasurableSpace β inst✝¹⁴ : TopologicalSpace γ inst✝¹³ : MeasurableSpace γ inst✝¹² : BorelSpace γ inst✝¹¹ : TopologicalSpace γ₂ inst✝¹⁰ : MeasurableSpace γ₂ inst✝⁹ : BorelSpace γ₂ inst✝⁸ : MeasurableSpace δ α' : Type u_6 inst✝⁷ : TopologicalSpace α' inst✝⁶ : MeasurableSpace α' inst✝⁵ : LinearOrder α✝ inst✝⁴ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α hd : Dense s hbot : ∀ (x : α), IsTop x → x ∈ s hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ e_1✝ : α = αᵒᵈ l : α hl : l ∈ s u : α hu : u ∈ s hlt : l < u ⊢ Ioc l u ∈ {S | ∃ l, l ∈ ↑OrderDual.ofDual ⁻¹' s ∧ ∃ u, u ∈ ↑OrderDual.ofDual ⁻¹' s ∧ l < u ∧ Ico l u = S} case h.e'_3.h.h.e'_2.h.h.mpr.intro.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α✝ inst✝²⁰ : TopologicalSpace α✝ inst✝¹⁹ : MeasurableSpace α✝ inst✝¹⁸ : OpensMeasurableSpace α✝ inst✝¹⁷ : TopologicalSpace β inst✝¹⁶ : MeasurableSpace β inst✝¹⁵ : OpensMeasurableSpace β inst✝¹⁴ : TopologicalSpace γ inst✝¹³ : MeasurableSpace γ inst✝¹² : BorelSpace γ inst✝¹¹ : TopologicalSpace γ₂ inst✝¹⁰ : MeasurableSpace γ₂ inst✝⁹ : BorelSpace γ₂ inst✝⁸ : MeasurableSpace δ α' : Type u_6 inst✝⁷ : TopologicalSpace α' inst✝⁶ : MeasurableSpace α' inst✝⁵ : LinearOrder α✝ inst✝⁴ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α hd : Dense s hbot : ∀ (x : α), IsTop x → x ∈ s hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s e_1✝¹ : MeasurableSpace α = MeasurableSpace αᵒᵈ e_1✝ : α = αᵒᵈ l : αᵒᵈ hl : l ∈ ↑OrderDual.ofDual ⁻¹' s u : αᵒᵈ hu : u ∈ ↑OrderDual.ofDual ⁻¹' s hlt : l < u ⊢ Ico l u ∈ {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ioc l u = S} ** exacts [⟨u, hu, l, hl, hlt, dual_Ico⟩, ⟨u, hu, l, hl, hlt, dual_Ioc⟩] ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²⁰ : TopologicalSpace α✝ inst✝¹⁹ : MeasurableSpace α✝ inst✝¹⁸ : OpensMeasurableSpace α✝ inst✝¹⁷ : TopologicalSpace β inst✝¹⁶ : MeasurableSpace β inst✝¹⁵ : OpensMeasurableSpace β inst✝¹⁴ : TopologicalSpace γ inst✝¹³ : MeasurableSpace γ inst✝¹² : BorelSpace γ inst✝¹¹ : TopologicalSpace γ₂ inst✝¹⁰ : MeasurableSpace γ₂ inst✝⁹ : BorelSpace γ₂ inst✝⁸ : MeasurableSpace δ α' : Type u_6 inst✝⁷ : TopologicalSpace α' inst✝⁶ : MeasurableSpace α' inst✝⁵ : LinearOrder α✝ inst✝⁴ : OrderClosedTopology α✝ a b x✝ : α✝ α : Type u_7 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α hd : Dense s hbot : ∀ (x : α), IsTop x → x ∈ s hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s x y : αᵒᵈ hlt : x < y he : Ioo x y = ∅ ⊢ Ioo y x = ∅ ** erw [dual_Ioo] ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²⁰ : TopologicalSpace α✝ inst✝¹⁹ : MeasurableSpace α✝ inst✝¹⁸ : OpensMeasurableSpace α✝ inst✝¹⁷ : TopologicalSpace β inst✝¹⁶ : MeasurableSpace β inst✝¹⁵ : OpensMeasurableSpace β inst✝¹⁴ : TopologicalSpace γ inst✝¹³ : MeasurableSpace γ inst✝¹² : BorelSpace γ inst✝¹¹ : TopologicalSpace γ₂ inst✝¹⁰ : MeasurableSpace γ₂ inst✝⁹ : BorelSpace γ₂ inst✝⁸ : MeasurableSpace δ α' : Type u_6 inst✝⁷ : TopologicalSpace α' inst✝⁶ : MeasurableSpace α' inst✝⁵ : LinearOrder α✝ inst✝⁴ : OrderClosedTopology α✝ a b x✝ : α✝ α : Type u_7 inst✝³ : TopologicalSpace α inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α hd : Dense s hbot : ∀ (x : α), IsTop x → x ∈ s hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s x y : αᵒᵈ hlt : x < y he : Ioo x y = ∅ ⊢ ↑OrderDual.ofDual ⁻¹' Ioo x y = ∅ ** exact he ** Qed
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Dense.borel_eq_generateFrom_Ioc_mem ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α inst✝⁴ : LinearOrder α inst✝³ : OrderTopology α inst✝² : SecondCountableTopology α inst✝¹ : DenselyOrdered α inst✝ : NoMaxOrder α s : Set α hd : Dense s ⊢ ∀ (x : α), IsTop x → x ∈ s ** simp ** Qed
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MeasureTheory.Measure.ext_of_Ico' ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) ⊢ μ = ν ** rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, hsb, _⟩ ** case intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s ⊢ μ = ν ** have : (⋃ (l ∈ s) (u ∈ s) (_ : l < u), {Ico l u} : Set (Set α)).Countable :=
hsc.biUnion fun l _ => hsc.biUnion fun u _ => countable_iUnion fun _ => countable_singleton _ ** case intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable (⋃ l ∈ s, ⋃ u ∈ s, ⋃ (_ : l < u), {Ico l u}) ⊢ μ = ν ** simp only [← setOf_eq_eq_singleton, ← setOf_exists] at this ** case intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊢ μ = ν ** refine'
Measure.ext_of_generateFrom_of_cover_subset
(BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico α)) (isPiSystem_Ico id id) _ this
_ _ _ ** case intro.intro.intro.intro.refine'_1 α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊢ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊆ {S | ∃ l u, l < u ∧ Ico l u = S} ** rintro _ ⟨l, -, u, -, h, rfl⟩ ** case intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h✝ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} l u : α h : l < u ⊢ Ico l u ∈ {S | ∃ l u, l < u ∧ Ico l u = S} ** exact ⟨l, u, h, rfl⟩ ** case intro.intro.intro.intro.refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊢ ⋃₀ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} = univ ** refine' sUnion_eq_univ_iff.2 fun x => _ ** case intro.intro.intro.intro.refine'_2 α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x✝ : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} x : α ⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b ** rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩ ** case intro.intro.intro.intro.refine'_2.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x✝ : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} x l : α hls : l ∈ s hlx : l ≤ x ⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b ** rcases hsd.exists_gt x with ⟨u, hus, hxu⟩ ** case intro.intro.intro.intro.refine'_2.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x✝ : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} x l : α hls : l ∈ s hlx : l ≤ x u : α hus : u ∈ s hxu : x < u ⊢ ∃ b, b ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ∧ x ∈ b ** exact ⟨_, ⟨l, hls, u, hus, hlx.trans_lt hxu, rfl⟩, hlx, hxu⟩ ** case intro.intro.intro.intro.refine'_3 α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊢ ∀ (s_1 : Set α), s_1 ∈ {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} → ↑↑μ s_1 ≠ ⊤ ** rintro _ ⟨l, -, u, -, hlt, rfl⟩ ** case intro.intro.intro.intro.refine'_3.intro.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} l u : α hlt : l < u ⊢ ↑↑μ (Ico l u) ≠ ⊤ ** exact hμ hlt ** case intro.intro.intro.intro.refine'_4 α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} ⊢ ∀ (s : Set α), s ∈ {S | ∃ l u, l < u ∧ Ico l u = S} → ↑↑μ s = ↑↑ν s ** rintro _ ⟨l, u, hlt, rfl⟩ ** case intro.intro.intro.intro.refine'_4.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α m : MeasurableSpace α inst✝⁴ : SecondCountableTopology α inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : BorelSpace α inst✝ : NoMaxOrder α μ ν : Measure α hμ : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) ≠ ⊤ h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b) s : Set α hsc : Set.Countable s hsd : Dense s hsb : ∀ (x : α), IsBot x → x ∈ s right✝ : ∀ (x : α), IsTop x → x ∈ s this : Set.Countable {x | ∃ i i_1 i_2 i_3 i_4, x = Ico i i_2} l u : α hlt : l < u ⊢ ↑↑μ (Ico l u) = ↑↑ν (Ico l u) ** exact h hlt ** Qed
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Measurable.max ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹⁷ : TopologicalSpace α inst✝¹⁶ : MeasurableSpace α inst✝¹⁵ : OpensMeasurableSpace α inst✝¹⁴ : TopologicalSpace β inst✝¹³ : MeasurableSpace β inst✝¹² : OpensMeasurableSpace β inst✝¹¹ : TopologicalSpace γ inst✝¹⁰ : MeasurableSpace γ inst✝⁹ : BorelSpace γ inst✝⁸ : TopologicalSpace γ₂ inst✝⁷ : MeasurableSpace γ₂ inst✝⁶ : BorelSpace γ₂ inst✝⁵ : MeasurableSpace δ α' : Type u_6 inst✝⁴ : TopologicalSpace α' inst✝³ : MeasurableSpace α' inst✝² : LinearOrder α inst✝¹ : OrderClosedTopology α a b : α inst✝ : SecondCountableTopology α f g : δ → α hf : Measurable f hg : Measurable g ⊢ Measurable fun a => Max.max (f a) (g a) ** simpa only [max_def'] using hf.piecewise (measurableSet_le hg hf) hg ** Qed
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Measurable.min ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹⁷ : TopologicalSpace α inst✝¹⁶ : MeasurableSpace α inst✝¹⁵ : OpensMeasurableSpace α inst✝¹⁴ : TopologicalSpace β inst✝¹³ : MeasurableSpace β inst✝¹² : OpensMeasurableSpace β inst✝¹¹ : TopologicalSpace γ inst✝¹⁰ : MeasurableSpace γ inst✝⁹ : BorelSpace γ inst✝⁸ : TopologicalSpace γ₂ inst✝⁷ : MeasurableSpace γ₂ inst✝⁶ : BorelSpace γ₂ inst✝⁵ : MeasurableSpace δ α' : Type u_6 inst✝⁴ : TopologicalSpace α' inst✝³ : MeasurableSpace α' inst✝² : LinearOrder α inst✝¹ : OrderClosedTopology α a b : α inst✝ : SecondCountableTopology α f g : δ → α hf : Measurable f hg : Measurable g ⊢ Measurable fun a => Min.min (f a) (g a) ** simpa only [min_def] using hf.piecewise (measurableSet_le hf hg) hg ** Qed
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ContinuousOn.measurable_piecewise ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s ⊢ Measurable (Set.piecewise s f g) ** refine' measurable_of_isOpen fun t ht => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t ⊢ MeasurableSet (Set.piecewise s f g ⁻¹' t) ** rw [piecewise_preimage, Set.ite] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t ⊢ MeasurableSet (f ⁻¹' t ∩ s ∪ g ⁻¹' t \ s) ** apply MeasurableSet.union ** case h₁ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t ⊢ MeasurableSet (f ⁻¹' t ∩ s) ** rcases _root_.continuousOn_iff'.1 hf t ht with ⟨u, u_open, hu⟩ ** case h₁.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u✝ : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t u : Set α u_open : IsOpen u hu : f ⁻¹' t ∩ s = u ∩ s ⊢ MeasurableSet (f ⁻¹' t ∩ s) ** rw [hu] ** case h₁.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u✝ : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t u : Set α u_open : IsOpen u hu : f ⁻¹' t ∩ s = u ∩ s ⊢ MeasurableSet (u ∩ s) ** exact u_open.measurableSet.inter hs ** case h₂ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t ⊢ MeasurableSet (g ⁻¹' t \ s) ** rcases _root_.continuousOn_iff'.1 hg t ht with ⟨u, u_open, hu⟩ ** case h₂.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u✝ : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t u : Set α u_open : IsOpen u hu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ ⊢ MeasurableSet (g ⁻¹' t \ s) ** rw [diff_eq_compl_inter, inter_comm, hu] ** case h₂.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u✝ : Set α inst✝¹⁵ : TopologicalSpace α inst✝¹⁴ : MeasurableSpace α inst✝¹³ : OpensMeasurableSpace α inst✝¹² : TopologicalSpace β inst✝¹¹ : MeasurableSpace β inst✝¹⁰ : OpensMeasurableSpace β inst✝⁹ : TopologicalSpace γ inst✝⁸ : MeasurableSpace γ inst✝⁷ : BorelSpace γ inst✝⁶ : TopologicalSpace γ₂ inst✝⁵ : MeasurableSpace γ₂ inst✝⁴ : BorelSpace γ₂ inst✝³ : MeasurableSpace δ α' : Type u_6 inst✝² : TopologicalSpace α' inst✝¹ : MeasurableSpace α' f g : α → γ s : Set α inst✝ : (j : α) → Decidable (j ∈ s) hf : ContinuousOn f s hg : ContinuousOn g sᶜ hs : MeasurableSet s t : Set γ ht : IsOpen t u : Set α u_open : IsOpen u hu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ ⊢ MeasurableSet (u ∩ sᶜ) ** exact u_open.measurableSet.inter hs.compl ** Qed
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prod_le_borel_prod ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝⁹ : TopologicalSpace α inst✝⁸ : MeasurableSpace α inst✝⁷ : BorelSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : MeasurableSpace β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : MeasurableSpace γ inst✝¹ : BorelSpace γ inst✝ : MeasurableSpace δ ⊢ Prod.instMeasurableSpace ≤ borel (α × β) ** refine' sup_le _ _ ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝⁹ : TopologicalSpace α inst✝⁸ : MeasurableSpace α inst✝⁷ : BorelSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : MeasurableSpace β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : MeasurableSpace γ inst✝¹ : BorelSpace γ inst✝ : MeasurableSpace δ ⊢ MeasurableSpace.comap Prod.fst (borel α) ≤ borel (α × β) ** exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝⁹ : TopologicalSpace α inst✝⁸ : MeasurableSpace α inst✝⁷ : BorelSpace α inst✝⁶ : TopologicalSpace β inst✝⁵ : MeasurableSpace β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : MeasurableSpace γ inst✝¹ : BorelSpace γ inst✝ : MeasurableSpace δ ⊢ MeasurableSpace.comap Prod.snd (borel β) ≤ borel (α × β) ** exact comap_le_iff_le_map.mpr continuous_snd.borel_measurable ** Qed
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measurable_of_Iio ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x) ⊢ Measurable f ** convert measurable_generateFrom (α := δ) _ ** case h.e'_4 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x) ⊢ inst✝¹¹ = MeasurableSpace.generateFrom ?convert_2 case convert_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x) ⊢ Set (Set α) case convert_4 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x) ⊢ ∀ (t : Set α), t ∈ ?convert_2 → MeasurableSet (f ⁻¹' t) ** exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Iio _) ** case convert_4 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x) ⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet (f ⁻¹' t) ** rintro _ ⟨x, rfl⟩ ** case convert_4.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iio x) x : α ⊢ MeasurableSet (f ⁻¹' Iio x) ** exact hf x ** Qed
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measurable_of_Ioi ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x) ⊢ Measurable f ** convert measurable_generateFrom (α := δ) _ ** case h.e'_4 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x) ⊢ inst✝¹¹ = MeasurableSpace.generateFrom ?convert_2 case convert_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x) ⊢ Set (Set α) case convert_4 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x) ⊢ ∀ (t : Set α), t ∈ ?convert_2 → MeasurableSet (f ⁻¹' t) ** exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioi _) ** case convert_4 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x) ⊢ ∀ (t : Set α), t ∈ range Ioi → MeasurableSet (f ⁻¹' t) ** rintro _ ⟨x, rfl⟩ ** case convert_4.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x) x : α ⊢ MeasurableSet (f ⁻¹' Ioi x) ** exact hf x ** Qed
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measurable_of_Iic ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x) ⊢ Measurable f ** apply measurable_of_Ioi ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x) ⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x) ** simp_rw [← compl_Iic, preimage_compl, MeasurableSet.compl_iff] ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Iic x) ⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iic x) ** assumption ** Qed
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measurable_of_Ici ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x) ⊢ Measurable f ** apply measurable_of_Iio ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x) ⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Iio x) ** simp_rw [← compl_Ici, preimage_compl, MeasurableSet.compl_iff] ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α f : δ → α hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x) ⊢ ∀ (x : α), MeasurableSet (f ⁻¹' Ici x) ** assumption ** Qed
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Measurable.isLUB ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g : δ → α hf : ∀ (i : ι), Measurable (f i) hg : ∀ (b : δ), IsLUB {a | ∃ i, f i b = a} (g b) ⊢ Measurable g ** change ∀ b, IsLUB (range fun i => f i b) (g b) at hg ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g : δ → α hf : ∀ (i : ι), Measurable (f i) hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b) ⊢ Measurable g ** apply measurable_generateFrom ** case h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g : δ → α hf : ∀ (i : ι), Measurable (f i) hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b) ⊢ ∀ (t : Set α), t ∈ range Ioi → MeasurableSet (g ⁻¹' t) ** rintro _ ⟨a, rfl⟩ ** case h.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g : δ → α hf : ∀ (i : ι), Measurable (f i) hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b) a : α ⊢ MeasurableSet (g ⁻¹' Ioi a) ** simp_rw [Set.preimage, mem_Ioi, lt_isLUB_iff (hg _), exists_range_iff, setOf_exists] ** case h.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g : δ → α hf : ∀ (i : ι), Measurable (f i) hg : ∀ (b : δ), IsLUB (range fun i => f i b) (g b) a : α ⊢ MeasurableSet (⋃ i, {x | a < f i x}) ** exact MeasurableSet.iUnion fun i => hf i (isOpen_lt' _).measurableSet ** Qed
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Measurable.isLUB_of_mem ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' ⊢ Measurable g ** rcases isEmpty_or_nonempty ι with hι|⟨⟨i⟩⟩ ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι ⊢ Measurable g ** rcases eq_empty_or_nonempty s with rfl|⟨x, hx⟩ ** case inl.inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) g'_meas : Measurable g' hι : IsEmpty ι hs : MeasurableSet ∅ hg : ∀ (b : δ), b ∈ ∅ → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' ∅ᶜ ⊢ Measurable g ** convert g'_meas ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) g'_meas : Measurable g' hι : IsEmpty ι hs : MeasurableSet ∅ hg : ∀ (b : δ), b ∈ ∅ → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' ∅ᶜ ⊢ g = g' ** ext x ** case h.e'_5.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) g'_meas : Measurable g' hι : IsEmpty ι hs : MeasurableSet ∅ hg : ∀ (b : δ), b ∈ ∅ → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' ∅ᶜ x : δ ⊢ g x = g' x ** simp only [compl_empty] at hg' ** case h.e'_5.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) g'_meas : Measurable g' hι : IsEmpty ι hs : MeasurableSet ∅ hg : ∀ (b : δ), b ∈ ∅ → IsLUB {a | ∃ i, f i b = a} (g b) x : δ hg' : EqOn g g' univ ⊢ g x = g' x ** exact hg' (mem_univ x) ** case inl.inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s ⊢ Measurable g ** have A : ∀ b ∈ s, IsBot (g b) := by simpa using hg ** case inl.inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) ⊢ Measurable g ** have B : ∀ b ∈ s, g b = g x := by
intro b hb
apply le_antisymm (A b hb (g x)) (A x hx (g b)) ** case inl.inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) B : ∀ (b : δ), b ∈ s → g b = g x this : g = piecewise s (fun _y => g x) g' ⊢ Measurable g ** rw [this] ** case inl.inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) B : ∀ (b : δ), b ∈ s → g b = g x this : g = piecewise s (fun _y => g x) g' ⊢ Measurable (piecewise s (fun _y => g x) g') ** exact Measurable.piecewise hs measurable_const g'_meas ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s ⊢ ∀ (b : δ), b ∈ s → IsBot (g b) ** simpa using hg ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) ⊢ ∀ (b : δ), b ∈ s → g b = g x ** intro b hb ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) b : δ hb : b ∈ s ⊢ g b = g x ** apply le_antisymm (A b hb (g x)) (A x hx (g b)) ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) B : ∀ (b : δ), b ∈ s → g b = g x ⊢ g = piecewise s (fun _y => g x) g' ** ext b ** case h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) B : ∀ (b : δ), b ∈ s → g b = g x b : δ ⊢ g b = piecewise s (fun _y => g x) g' b ** by_cases hb : b ∈ s ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) B : ∀ (b : δ), b ∈ s → g b = g x b : δ hb : b ∈ s ⊢ g b = piecewise s (fun _y => g x) g' b ** simp [hb, B] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' hι : IsEmpty ι x : δ hx : x ∈ s A : ∀ (b : δ), b ∈ s → IsBot (g b) B : ∀ (b : δ), b ∈ s → g b = g x b : δ hb : ¬b ∈ s ⊢ g b = piecewise s (fun _y => g x) g' b ** simp [hb, hg' hb] ** case inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι ⊢ Measurable g ** let f' : ι → δ → α := fun i ↦ s.piecewise (f i) g' ** case inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' ⊢ Measurable g ** suffices ∀ b, IsLUB { a | ∃ i, f' i b = a } (g b) from
Measurable.isLUB (fun i ↦ Measurable.piecewise hs (hf i) g'_meas) this ** case inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' ⊢ ∀ (b : δ), IsLUB {a | ∃ i, f' i b = a} (g b) ** intro b ** case inr.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ ⊢ IsLUB {a | ∃ i, f' i b = a} (g b) ** by_cases hb : b ∈ s ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : b ∈ s ⊢ IsLUB {a | ∃ i, f' i b = a} (g b) ** have A : ∀ i, f' i b = f i b := fun i ↦ by simp [hb] ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : b ∈ s A : ∀ (i : ι), f' i b = f i b ⊢ IsLUB {a | ∃ i, f' i b = a} (g b) ** simpa [A] using hg b hb ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i✝ : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : b ∈ s i : ι ⊢ f' i b = f i b ** simp [hb] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s ⊢ IsLUB {a | ∃ i, f' i b = a} (g b) ** have A : ∀ i, f' i b = g' b := fun i ↦ by simp [hb] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s A : ∀ (i : ι), f' i b = g' b this : {a | ∃ _i, g' b = a} = {g' b} ⊢ IsLUB {a | ∃ i, f' i b = a} (g b) ** simp [A, this, hg' hb, isLUB_singleton] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i✝ : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s i : ι ⊢ f' i b = g' b ** simp [hb] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s A : ∀ (i : ι), f' i b = g' b ⊢ {a | ∃ _i, g' b = a} = {g' b} ** apply Subset.antisymm ** case h₁ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s A : ∀ (i : ι), f' i b = g' b ⊢ {a | ∃ _i, g' b = a} ⊆ {g' b} ** rintro - ⟨_j, rfl⟩ ** case h₁.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s A : ∀ (i : ι), f' i b = g' b _j : ι ⊢ g' b ∈ {g' b} ** simp only [mem_singleton_iff] ** case h₂ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s A : ∀ (i : ι), f' i b = g' b ⊢ {g' b} ⊆ {a | ∃ _i, g' b = a} ** rintro - rfl ** case h₂ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : LinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 inst✝ : Countable ι f : ι → δ → α g g' : δ → α hf : ∀ (i : ι), Measurable (f i) s : Set δ hs : MeasurableSet s hg : ∀ (b : δ), b ∈ s → IsLUB {a | ∃ i, f i b = a} (g b) hg' : EqOn g g' sᶜ g'_meas : Measurable g' i : ι f' : ι → δ → α := fun i => piecewise s (f i) g' b : δ hb : ¬b ∈ s A : ∀ (i : ι), f' i b = g' b ⊢ g' b ∈ {a | ∃ _i, g' b = a} ** exact ⟨i, rfl⟩ ** Qed
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measurableSet_of_mem_nhdsWithin_Ioi_aux ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x h' : ∀ (x : α), x ∈ s → ∃ y, x < y ⊢ MeasurableSet s ** choose! M hM using h' ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x ⊢ MeasurableSet s ** suffices H : (s \ interior s).Countable ** case H α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x ⊢ Set.Countable (s \ interior s) ** have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
(mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx) ** case H α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x A : ∀ (x : α), x ∈ s → ∃ y, y ∈ Ioi x ∧ Ioo x y ⊆ s ⊢ Set.Countable (s \ interior s) ** choose! y hy h'y using A ** case H α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s B : PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) ⊢ Set.Countable (s \ interior s) ** exact B.countable_of_Ioo fun x hx => hy x hx.1 ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x H : Set.Countable (s \ interior s) ⊢ MeasurableSet s ** have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x H : Set.Countable (s \ interior s) this : s = interior s ∪ s \ interior s ⊢ MeasurableSet s ** rw [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x H : Set.Countable (s \ interior s) this : s = interior s ∪ s \ interior s ⊢ MeasurableSet (interior s ∪ s \ interior s) ** exact isOpen_interior.measurableSet.union H.measurableSet ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x H : Set.Countable (s \ interior s) ⊢ s = interior s ∪ s \ interior s ** rw [union_diff_cancel interior_subset] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s ⊢ PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) ** intro x hx x' hx' hxx' ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s x : α hx : x ∈ s \ interior s x' : α hx' : x' ∈ s \ interior s hxx' : x ≠ x' ⊢ (Disjoint on fun x => Ioo x (y x)) x x' ** rcases lt_or_gt_of_ne hxx' with (h' | h') ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s x : α hx : x ∈ s \ interior s x' : α hx' : x' ∈ s \ interior s hxx' : x ≠ x' h' : x < x' ⊢ (Disjoint on fun x => Ioo x (y x)) x x' ** refine disjoint_left.2 fun z hz h'z => ?_ ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s x : α hx : x ∈ s \ interior s x' : α hx' : x' ∈ s \ interior s hxx' : x ≠ x' h' : x < x' z : α hz : z ∈ (fun x => Ioo x (y x)) x h'z : z ∈ (fun x => Ioo x (y x)) x' ⊢ False ** have : x' ∈ interior s :=
mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩ ** case inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s x : α hx : x ∈ s \ interior s x' : α hx' : x' ∈ s \ interior s hxx' : x ≠ x' h' : x < x' z : α hz : z ∈ (fun x => Ioo x (y x)) x h'z : z ∈ (fun x => Ioo x (y x)) x' this : x' ∈ interior s ⊢ False ** exact False.elim (hx'.2 this) ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s x : α hx : x ∈ s \ interior s x' : α hx' : x' ∈ s \ interior s hxx' : x ≠ x' h' : x > x' ⊢ (Disjoint on fun x => Ioo x (y x)) x x' ** refine disjoint_left.2 fun z hz h'z => ?_ ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s x : α hx : x ∈ s \ interior s x' : α hx' : x' ∈ s \ interior s hxx' : x ≠ x' h' : x > x' z : α hz : z ∈ (fun x => Ioo x (y x)) x h'z : z ∈ (fun x => Ioo x (y x)) x' ⊢ False ** have : x ∈ interior s :=
mem_interior.2 ⟨Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, ⟨h', hz.1.trans h'z.2⟩⟩ ** case inr α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x M : α → α hM : ∀ (x : α), x ∈ s → x < M x y : α → α hy : ∀ (x : α), x ∈ s → y x ∈ Ioi x h'y : ∀ (x : α), x ∈ s → Ioo x (y x) ⊆ s x : α hx : x ∈ s \ interior s x' : α hx' : x' ∈ s \ interior s hxx' : x ≠ x' h' : x > x' z : α hz : z ∈ (fun x => Ioo x (y x)) x h'z : z ∈ (fun x => Ioo x (y x)) x' this : x ∈ interior s ⊢ False ** exact False.elim (hx.2 this) ** Qed
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measurableSet_of_mem_nhdsWithin_Ioi ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x ⊢ MeasurableSet s ** by_cases H : ∃ x ∈ s, IsTop x ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x H : ∃ x, x ∈ s ∧ IsTop x ⊢ MeasurableSet s ** rcases H with ⟨x₀, x₀s, h₀⟩ ** case pos.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ ⊢ MeasurableSet s ** have : s = {x₀} ∪ s \ {x₀} := by rw [union_diff_cancel (singleton_subset_iff.2 x₀s)] ** case pos.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} ⊢ MeasurableSet s ** rw [this] ** case pos.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} ⊢ MeasurableSet ({x₀} ∪ s \ {x₀}) ** refine' (measurableSet_singleton _).union _ ** case pos.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} ⊢ MeasurableSet (s \ {x₀}) ** have A : ∀ x ∈ s \ {x₀}, x < x₀ := fun x hx => lt_of_le_of_ne (h₀ _) (by simpa using hx.2) ** case pos.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀ ⊢ MeasurableSet (s \ {x₀}) ** refine' measurableSet_of_mem_nhdsWithin_Ioi_aux (fun x hx => _) fun x hx => ⟨x₀, A x hx⟩ ** case pos.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀ x : α hx : x ∈ s \ {x₀} ⊢ s \ {x₀} ∈ 𝓝[Ioi x] x ** obtain ⟨u, hu, us⟩ : ∃ (u : α), u ∈ Ioi x ∧ Ioo x u ⊆ s :=
(mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).1 (h x hx.1) ** case pos.intro.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀ x : α hx : x ∈ s \ {x₀} u : α hu : u ∈ Ioi x us : Ioo x u ⊆ s ⊢ s \ {x₀} ∈ 𝓝[Ioi x] x ** refine' (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).2 ⟨u, hu, fun y hy => ⟨us hy, _⟩⟩ ** case pos.intro.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u✝ : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} A : ∀ (x : α), x ∈ s \ {x₀} → x < x₀ x : α hx : x ∈ s \ {x₀} u : α hu : u ∈ Ioi x us : Ioo x u ⊆ s y : α hy : y ∈ Ioo x u ⊢ ¬y ∈ {x₀} ** exact ne_of_lt (hy.2.trans_le (h₀ _)) ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ ⊢ s = {x₀} ∪ s \ {x₀} ** rw [union_diff_cancel (singleton_subset_iff.2 x₀s)] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x x₀ : α x₀s : x₀ ∈ s h₀ : IsTop x₀ this : s = {x₀} ∪ s \ {x₀} x : α hx : x ∈ s \ {x₀} ⊢ x ≠ x₀ ** simpa using hx.2 ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x H : ¬∃ x, x ∈ s ∧ IsTop x ⊢ MeasurableSet s ** apply measurableSet_of_mem_nhdsWithin_Ioi_aux h ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x H : ¬∃ x, x ∈ s ∧ IsTop x ⊢ ∀ (x : α), x ∈ s → ∃ y, x < y ** simp only [IsTop] at H ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x H : ¬∃ x, x ∈ s ∧ ∀ (b : α), b ≤ x ⊢ ∀ (x : α), x ∈ s → ∃ y, x < y ** push_neg at H ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : LinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α s : Set α h : ∀ (x : α), x ∈ s → s ∈ 𝓝[Ioi x] x H : ∀ (x : α), x ∈ s → ∃ b, x < b ⊢ ∀ (x : α), x ∈ s → ∃ y, x < y ** exact H ** Qed
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aemeasurable_iSup ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : ConditionallyCompleteLinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 μ : Measure δ inst✝ : Countable ι f : ι → δ → α hf : ∀ (i : ι), AEMeasurable (f i) ⊢ AEMeasurable fun b => ⨆ i, f i b ** refine ⟨fun b ↦ ⨆ i, (hf i).mk (f i) b, measurable_iSup (fun i ↦ (hf i).measurable_mk), ?_⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : ConditionallyCompleteLinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 μ : Measure δ inst✝ : Countable ι f : ι → δ → α hf : ∀ (i : ι), AEMeasurable (f i) ⊢ (fun b => ⨆ i, f i b) =ᵐ[μ] fun b => ⨆ i, AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) b ** filter_upwards [ae_all_iff.2 (fun i ↦ (hf i).ae_eq_mk)] with b hb using by simp [hb] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹³ : TopologicalSpace α inst✝¹² : MeasurableSpace α inst✝¹¹ : BorelSpace α inst✝¹⁰ : TopologicalSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : TopologicalSpace γ inst✝⁶ : MeasurableSpace γ inst✝⁵ : BorelSpace γ inst✝⁴ : MeasurableSpace δ inst✝³ : ConditionallyCompleteLinearOrder α inst✝² : OrderTopology α inst✝¹ : SecondCountableTopology α ι : Sort u_6 μ : Measure δ inst✝ : Countable ι f : ι → δ → α hf : ∀ (i : ι), AEMeasurable (f i) b : δ hb : ∀ (i : ι), f i b = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) b ⊢ ⨆ i, f i b = ⨆ i, AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) b ** simp [hb] ** Qed
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measurable_sSup ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 f : ι → δ → α s : Set ι hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → Measurable (f i) ⊢ Measurable fun x => sSup ((fun i => f i x) '' s) ** have : Countable ↑s := countable_coe_iff.2 hs ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 f : ι → δ → α s : Set ι hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → Measurable (f i) this : Countable ↑s ⊢ Measurable fun x => sSup ((fun i => f i x) '' s) ** convert measurable_iSup (f := (fun (i : s) ↦ f i)) (fun i ↦ hf i i.2) using 1 ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 f : ι → δ → α s : Set ι hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → Measurable (f i) this : Countable ↑s ⊢ (fun x => sSup ((fun i => f i x) '' s)) = fun b => ⨆ i, f (↑i) b ** ext b ** case h.e'_5.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 f : ι → δ → α s : Set ι hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → Measurable (f i) this : Countable ↑s b : δ ⊢ sSup ((fun i => f i b) '' s) = ⨆ i, f (↑i) b ** congr ** case h.e'_5.h.e_a α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 f : ι → δ → α s : Set ι hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → Measurable (f i) this : Countable ↑s b : δ ⊢ (fun i => f i b) '' s = range fun i => f (↑i) b ** exact image_eq_range (fun i ↦ f i b) s ** Qed
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aemeasurable_biSup ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) ⊢ AEMeasurable fun b => ⨆ i ∈ s, f i b ** let g : ι → δ → α := fun i ↦ if hi : i ∈ s then (hf i hi).mk (f i) else fun _b ↦ sSup ∅ ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ ⊢ AEMeasurable fun b => ⨆ i ∈ s, f i b ** have : ∀ i ∈ s, Measurable (g i) := by
intro i hi
simpa [hi] using (hf i hi).measurable_mk ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this : ∀ (i : ι), i ∈ s → Measurable (g i) ⊢ AEMeasurable fun b => ⨆ i ∈ s, f i b ** refine ⟨fun b ↦ ⨆ (i) (_ : i ∈ s), g i b, measurable_biSup s hs this, ?_⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this : ∀ (i : ι), i ∈ s → Measurable (g i) ⊢ (fun b => ⨆ i ∈ s, f i b) =ᵐ[μ] fun b => ⨆ i ∈ s, g i b ** have : ∀ i ∈ s, ∀ᵐ b ∂μ, f i b = g i b :=
fun i hi ↦ by simpa [hi] using (hf i hi).ae_eq_mk ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this✝ : ∀ (i : ι), i ∈ s → Measurable (g i) this : ∀ (i : ι), i ∈ s → ∀ᵐ (b : δ) ∂μ, f i b = g i b ⊢ (fun b => ⨆ i ∈ s, f i b) =ᵐ[μ] fun b => ⨆ i ∈ s, g i b ** filter_upwards [(ae_ball_iff hs).2 this] with b hb ** case h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this✝ : ∀ (i : ι), i ∈ s → Measurable (g i) this : ∀ (i : ι), i ∈ s → ∀ᵐ (b : δ) ∂μ, f i b = g i b b : δ hb : ∀ (i : ι), i ∈ s → f i b = g i b ⊢ ⨆ i ∈ s, f i b = ⨆ i ∈ s, g i b ** congr ** case h.e_s α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this✝ : ∀ (i : ι), i ∈ s → Measurable (g i) this : ∀ (i : ι), i ∈ s → ∀ᵐ (b : δ) ∂μ, f i b = g i b b : δ hb : ∀ (i : ι), i ∈ s → f i b = g i b ⊢ (fun i => ⨆ (_ : i ∈ s), f i b) = fun i => ⨆ (_ : i ∈ s), g i b ** ext i ** case h.e_s.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this✝ : ∀ (i : ι), i ∈ s → Measurable (g i) this : ∀ (i : ι), i ∈ s → ∀ᵐ (b : δ) ∂μ, f i b = g i b b : δ hb : ∀ (i : ι), i ∈ s → f i b = g i b i : ι ⊢ ⨆ (_ : i ∈ s), f i b = ⨆ (_ : i ∈ s), g i b ** congr ** case h.e_s.h.e_s α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this✝ : ∀ (i : ι), i ∈ s → Measurable (g i) this : ∀ (i : ι), i ∈ s → ∀ᵐ (b : δ) ∂μ, f i b = g i b b : δ hb : ∀ (i : ι), i ∈ s → f i b = g i b i : ι ⊢ (fun h => f i b) = fun x => g i b ** ext hi ** case h.e_s.h.e_s.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this✝ : ∀ (i : ι), i ∈ s → Measurable (g i) this : ∀ (i : ι), i ∈ s → ∀ᵐ (b : δ) ∂μ, f i b = g i b b : δ hb : ∀ (i : ι), i ∈ s → f i b = g i b i : ι hi : i ∈ s ⊢ f i b = g i b ** simp [hi, hb] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ ⊢ ∀ (i : ι), i ∈ s → Measurable (g i) ** intro i hi ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ i : ι hi : i ∈ s ⊢ Measurable (g i) ** simpa [hi] using (hf i hi).measurable_mk ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : BorelSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : BorelSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : MeasurableSpace δ inst✝² : ConditionallyCompleteLinearOrder α inst✝¹ : OrderTopology α inst✝ : SecondCountableTopology α ι : Type u_6 μ : Measure δ s : Set ι f : ι → δ → α hs : Set.Countable s hf : ∀ (i : ι), i ∈ s → AEMeasurable (f i) g : ι → δ → α := fun i => if hi : i ∈ s then AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) else fun _b => sSup ∅ this : ∀ (i : ι), i ∈ s → Measurable (g i) i : ι hi : i ∈ s ⊢ ∀ᵐ (b : δ) ∂μ, f i b = g i b ** simpa [hi] using (hf i hi).ae_eq_mk ** Qed
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IsFiniteMeasureOnCompacts.map ** α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α✝ inst✝¹⁶ : TopologicalSpace α✝ inst✝¹⁵ : MeasurableSpace α✝ inst✝¹⁴ : BorelSpace α✝ inst✝¹³ : TopologicalSpace β✝ inst✝¹² : MeasurableSpace β✝ inst✝¹¹ : BorelSpace β✝ inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : MeasurableSpace δ α : Type u_6 m0 : MeasurableSpace α inst✝⁶ : TopologicalSpace α inst✝⁵ : OpensMeasurableSpace α β : Type u_7 inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : BorelSpace β inst✝¹ : T2Space β μ : Measure α inst✝ : IsFiniteMeasureOnCompacts μ f : α ≃ₜ β ⊢ ∀ ⦃K : Set β⦄, IsCompact K → ↑↑(Measure.map (↑f) μ) K < ⊤ ** intro K hK ** α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α✝ inst✝¹⁶ : TopologicalSpace α✝ inst✝¹⁵ : MeasurableSpace α✝ inst✝¹⁴ : BorelSpace α✝ inst✝¹³ : TopologicalSpace β✝ inst✝¹² : MeasurableSpace β✝ inst✝¹¹ : BorelSpace β✝ inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : MeasurableSpace δ α : Type u_6 m0 : MeasurableSpace α inst✝⁶ : TopologicalSpace α inst✝⁵ : OpensMeasurableSpace α β : Type u_7 inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : BorelSpace β inst✝¹ : T2Space β μ : Measure α inst✝ : IsFiniteMeasureOnCompacts μ f : α ≃ₜ β K : Set β hK : IsCompact K ⊢ ↑↑(Measure.map (↑f) μ) K < ⊤ ** rw [Measure.map_apply f.measurable hK.measurableSet] ** α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α✝ inst✝¹⁶ : TopologicalSpace α✝ inst✝¹⁵ : MeasurableSpace α✝ inst✝¹⁴ : BorelSpace α✝ inst✝¹³ : TopologicalSpace β✝ inst✝¹² : MeasurableSpace β✝ inst✝¹¹ : BorelSpace β✝ inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : MeasurableSpace δ α : Type u_6 m0 : MeasurableSpace α inst✝⁶ : TopologicalSpace α inst✝⁵ : OpensMeasurableSpace α β : Type u_7 inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : BorelSpace β inst✝¹ : T2Space β μ : Measure α inst✝ : IsFiniteMeasureOnCompacts μ f : α ≃ₜ β K : Set β hK : IsCompact K ⊢ ↑↑μ (↑f ⁻¹' K) < ⊤ ** apply IsCompact.measure_lt_top ** case hK α✝ : Type u_1 β✝ : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α✝ inst✝¹⁶ : TopologicalSpace α✝ inst✝¹⁵ : MeasurableSpace α✝ inst✝¹⁴ : BorelSpace α✝ inst✝¹³ : TopologicalSpace β✝ inst✝¹² : MeasurableSpace β✝ inst✝¹¹ : BorelSpace β✝ inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : MeasurableSpace δ α : Type u_6 m0 : MeasurableSpace α inst✝⁶ : TopologicalSpace α inst✝⁵ : OpensMeasurableSpace α β : Type u_7 inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : BorelSpace β inst✝¹ : T2Space β μ : Measure α inst✝ : IsFiniteMeasureOnCompacts μ f : α ≃ₜ β K : Set β hK : IsCompact K ⊢ IsCompact (↑f ⁻¹' K) ** rwa [f.isCompact_preimage] ** Qed
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tendsto_measure_thickening ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝³ : PseudoEMetricSpace α inst✝² : MeasurableSpace α inst✝¹ : OpensMeasurableSpace α inst✝ : MeasurableSpace β x : α ε : ℝ≥0∞ μ : Measure α s : Set α hs : ∃ R, R > 0 ∧ ↑↑μ (thickening R s) ≠ ⊤ ⊢ Tendsto (fun r => ↑↑μ (thickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closure s))) ** rw [closure_eq_iInter_thickening] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝³ : PseudoEMetricSpace α inst✝² : MeasurableSpace α inst✝¹ : OpensMeasurableSpace α inst✝ : MeasurableSpace β x : α ε : ℝ≥0∞ μ : Measure α s : Set α hs : ∃ R, R > 0 ∧ ↑↑μ (thickening R s) ≠ ⊤ ⊢ Tendsto (fun r => ↑↑μ (thickening r s)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (⋂ δ, ⋂ (_ : 0 < δ), thickening δ s))) ** exact tendsto_measure_biInter_gt (fun r _ => isOpen_thickening.measurableSet)
(fun i j _ ij => thickening_mono ij _) hs ** Qed
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Real.isPiSystem_Iio_rat ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ IsPiSystem (⋃ a, {Iio ↑a}) ** convert isPiSystem_image_Iio (((↑) : ℚ → ℝ) '' univ) ** case h.e'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ ⋃ a, {Iio ↑a} = Iio '' (Rat.cast '' univ) ** ext x ** case h.e'_2.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α x : Set ℝ ⊢ x ∈ ⋃ a, {Iio ↑a} ↔ x ∈ Iio '' (Rat.cast '' univ) ** simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and] ** Qed
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Real.isPiSystem_Ioi_rat ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ IsPiSystem (⋃ a, {Ioi ↑a}) ** convert isPiSystem_image_Ioi (((↑) : ℚ → ℝ) '' univ) ** case h.e'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ ⋃ a, {Ioi ↑a} = Ioi '' (Rat.cast '' univ) ** ext x ** case h.e'_2.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α x : Set ℝ ⊢ x ∈ ⋃ a, {Ioi ↑a} ↔ x ∈ Ioi '' (Rat.cast '' univ) ** simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and] ** Qed
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Real.isPiSystem_Iic_rat ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ IsPiSystem (⋃ a, {Iic ↑a}) ** convert isPiSystem_image_Iic (((↑) : ℚ → ℝ) '' univ) ** case h.e'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ ⋃ a, {Iic ↑a} = Iic '' (Rat.cast '' univ) ** ext x ** case h.e'_2.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α x : Set ℝ ⊢ x ∈ ⋃ a, {Iic ↑a} ↔ x ∈ Iic '' (Rat.cast '' univ) ** simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and] ** Qed
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Real.isPiSystem_Ici_rat ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ IsPiSystem (⋃ a, {Ici ↑a}) ** convert isPiSystem_image_Ici (((↑) : ℚ → ℝ) '' univ) ** case h.e'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α ⊢ ⋃ a, {Ici ↑a} = Ici '' (Rat.cast '' univ) ** ext x ** case h.e'_2.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α x : Set ℝ ⊢ x ∈ ⋃ a, {Ici ↑a} ↔ x ∈ Ici '' (Rat.cast '' univ) ** simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and] ** Qed
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aEMeasurable_coe_nnreal_real_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝ : MeasurableSpace α f : α → ℝ≥0 μ : Measure α h : AEMeasurable fun x => ↑(f x) ⊢ AEMeasurable f ** simpa only [Real.toNNReal_coe] using h.real_toNNReal ** Qed
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Measurable.ennreal_tsum ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ h : ∀ (i : ι), Measurable (f i) ⊢ Measurable fun x => ∑' (i : ι), f i x ** simp_rw [ENNReal.tsum_eq_iSup_sum] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ h : ∀ (i : ι), Measurable (f i) ⊢ Measurable fun x => ⨆ s, ∑ i in s, f i x ** apply measurable_iSup ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ h : ∀ (i : ι), Measurable (f i) ⊢ ∀ (i : Finset ι), Measurable fun b => ∑ i in i, f i b ** exact fun s => s.measurable_sum fun i _ => h i ** Qed
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Measurable.ennreal_tsum' ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ h : ∀ (i : ι), Measurable (f i) ⊢ Measurable (∑' (i : ι), f i) ** convert Measurable.ennreal_tsum h with x ** case h.e'_5.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ h : ∀ (i : ι), Measurable (f i) x : α ⊢ tsum (fun i => f i) x = ∑' (i : ι), f i x ** exact tsum_apply (Pi.summable.2 fun _ => ENNReal.summable) ** Qed
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AEMeasurable.ennreal_tsum ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ μ : Measure α h : ∀ (i : ι), AEMeasurable (f i) ⊢ AEMeasurable fun x => ∑' (i : ι), f i x ** simp_rw [ENNReal.tsum_eq_iSup_sum] ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ μ : Measure α h : ∀ (i : ι), AEMeasurable (f i) ⊢ AEMeasurable fun x => ⨆ s, ∑ i in s, f i x ** apply aemeasurable_iSup ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι✝ : Sort y s t u : Set α inst✝¹ : MeasurableSpace α ι : Type u_6 inst✝ : Countable ι f : ι → α → ℝ≥0∞ μ : Measure α h : ∀ (i : ι), AEMeasurable (f i) ⊢ ∀ (i : Finset ι), AEMeasurable fun b => ∑ i in i, f i b ** exact fun s => Finset.aemeasurable_sum s fun i _ => h i ** Qed
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EReal.measurable_of_measurable_real ** α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝ : MeasurableSpace α f : EReal → α h : Measurable fun p => f ↑p ⊢ Set.Finite {⊥, ⊤} ** simp ** Qed
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MeasureTheory.Egorov.mem_notConvergentSeq_iff ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝¹ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝ : Preorder ι x : α ⊢ x ∈ notConvergentSeq f g n j ↔ ∃ k x_1, 1 / (↑n + 1) < dist (f k x) (g x) ** simp_rw [notConvergentSeq, Set.mem_iUnion] ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝¹ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝ : Preorder ι x : α ⊢ (∃ i i_1, x ∈ {x | 1 / (↑n + 1) < dist (f i x) (g x)}) ↔ ∃ k x_1, 1 / (↑n + 1) < dist (f k x) (g x) ** rfl ** Qed
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MeasureTheory.Egorov.measure_inter_notConvergentSeq_eq_zero ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) n : ℕ ⊢ ↑↑μ (s ∩ ⋂ j, notConvergentSeq f g n j) = 0 ** simp_rw [Metric.tendsto_atTop, ae_iff] at hfg ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 ⊢ ↑↑μ (s ∩ ⋂ j, notConvergentSeq f g n j) = 0 ** rw [← nonpos_iff_eq_zero, ← hfg] ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 ⊢ ↑↑μ (s ∩ ⋂ j, notConvergentSeq f g n j) ≤ ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} ** refine' measure_mono fun x => _ ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 x : α ⊢ x ∈ s ∩ ⋂ j, notConvergentSeq f g n j → x ∈ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} ** simp only [Set.mem_inter_iff, Set.mem_iInter, ge_iff_le, mem_notConvergentSeq_iff] ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 x : α ⊢ (x ∈ s ∧ ∀ (i : ι), ∃ k x_1, 1 / (↑n + 1) < dist (f k x) (g x)) → x ∈ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), N ≤ n → dist (f n a) (g a) < ε)} ** push_neg ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 x : α ⊢ (x ∈ s ∧ ∀ (i : ι), ∃ k x_1, 1 / (↑n + 1) < dist (f k x) (g x)) → x ∈ {a | a ∈ s ∧ ∃ ε, ε > 0 ∧ ∀ (N : ι), ∃ n, N ≤ n ∧ ε ≤ dist (f n a) (g a)} ** rintro ⟨hmem, hx⟩ ** case intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 x : α hmem : x ∈ s hx : ∀ (i : ι), ∃ k x_1, 1 / (↑n + 1) < dist (f k x) (g x) ⊢ x ∈ {a | a ∈ s ∧ ∃ ε, ε > 0 ∧ ∀ (N : ι), ∃ n, N ≤ n ∧ ε ≤ dist (f n a) (g a)} ** refine' ⟨hmem, 1 / (n + 1 : ℝ), Nat.one_div_pos_of_nat, fun N => _⟩ ** case intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 x : α hmem : x ∈ s hx : ∀ (i : ι), ∃ k x_1, 1 / (↑n + 1) < dist (f k x) (g x) N : ι ⊢ ∃ n_1, N ≤ n_1 ∧ 1 / (↑n + 1) ≤ dist (f n_1 x) (g x) ** obtain ⟨n, hn₁, hn₂⟩ := hx N ** case intro.intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝² : MetricSpace β μ : Measure α n✝¹ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝¹ : SemilatticeSup ι inst✝ : Nonempty ι n✝ : ℕ hfg : ↑↑μ {a | ¬(a ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ N, ∀ (n : ι), n ≥ N → dist (f n a) (g a) < ε)} = 0 x : α hmem : x ∈ s hx : ∀ (i : ι), ∃ k x_1, 1 / (↑n✝ + 1) < dist (f k x) (g x) N n : ι hn₁ : N ≤ n hn₂ : 1 / (↑n✝ + 1) < dist (f n x) (g x) ⊢ ∃ n, N ≤ n ∧ 1 / (↑n✝ + 1) ≤ dist (f n x) (g x) ** exact ⟨n, hn₁, hn₂.le⟩ ** Qed
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MeasureTheory.Egorov.iUnionNotConvergentSeq_subset ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) ⊢ iUnionNotConvergentSeq hε hf hg hsm hs hfg ⊆ s ** rw [iUnionNotConvergentSeq, ← Set.inter_iUnion] ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) ⊢ s ∩ ⋃ i, notConvergentSeq (fun n => f n) g i (notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg i) ⊆ s ** exact Set.inter_subset_left _ _ ** Qed
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MeasureTheory.Egorov.tendstoUniformlyOn_diff_iUnionNotConvergentSeq ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) ⊢ TendstoUniformlyOn f g atTop (s \ iUnionNotConvergentSeq hε hf hg hsm hs hfg) ** rw [Metric.tendstoUniformlyOn_iff] ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) ⊢ ∀ (ε_1 : ℝ), ε_1 > 0 → ∀ᶠ (n : ι) in atTop, ∀ (x : α), x ∈ s \ iUnionNotConvergentSeq hε hf hg hsm hs hfg → dist (g x) (f n x) < ε_1 ** intro δ hδ ** α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 ⊢ ∀ᶠ (n : ι) in atTop, ∀ (x : α), x ∈ s \ iUnionNotConvergentSeq hε hf hg hsm hs hfg → dist (g x) (f n x) < δ ** obtain ⟨N, hN⟩ := exists_nat_one_div_lt hδ ** case intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ ⊢ ∀ᶠ (n : ι) in atTop, ∀ (x : α), x ∈ s \ iUnionNotConvergentSeq hε hf hg hsm hs hfg → dist (g x) (f n x) < δ ** rw [eventually_atTop] ** case intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ ⊢ ∃ a, ∀ (b : ι), b ≥ a → ∀ (x : α), x ∈ s \ iUnionNotConvergentSeq hε hf hg hsm hs hfg → dist (g x) (f b x) < δ ** refine' ⟨Egorov.notConvergentSeqLTIndex (half_pos hε) hf hg hsm hs hfg N, fun n hn x hx => _⟩ ** case intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ n : ι hn : n ≥ notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N x : α hx : x ∈ s \ iUnionNotConvergentSeq hε hf hg hsm hs hfg ⊢ dist (g x) (f n x) < δ ** simp only [Set.mem_diff, Egorov.iUnionNotConvergentSeq, not_exists, Set.mem_iUnion,
Set.mem_inter_iff, not_and, exists_and_left] at hx ** case intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ n : ι hn : n ≥ notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N x : α hx : x ∈ s ∧ (x ∈ s → ∀ (x_1 : ℕ), ¬x ∈ notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg x_1)) ⊢ dist (g x) (f n x) < δ ** obtain ⟨hxs, hx⟩ := hx ** case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ n : ι hn : n ≥ notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N x : α hxs : x ∈ s hx : x ∈ s → ∀ (x_1 : ℕ), ¬x ∈ notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg x_1) ⊢ dist (g x) (f n x) < δ ** specialize hx hxs N ** case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ n : ι hn : n ≥ notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N x : α hxs : x ∈ s hx : ¬x ∈ notConvergentSeq (fun n => f n) g N (notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N) ⊢ dist (g x) (f n x) < δ ** rw [Egorov.mem_notConvergentSeq_iff] at hx ** case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ n : ι hn : n ≥ notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N x : α hxs : x ∈ s hx : ¬∃ k x_1, 1 / (↑N + 1) < dist (f k x) (g x) ⊢ dist (g x) (f n x) < δ ** push_neg at hx ** case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ n : ι hn : n ≥ notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N x : α hxs : x ∈ s hx : ∀ (k : ι), notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N ≤ k → dist (f k x) (g x) ≤ 1 / (↑N + 1) ⊢ dist (g x) (f n x) < δ ** rw [dist_comm] ** case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace α inst✝³ : MetricSpace β μ : Measure α n✝ : ℕ i j : ι s : Set α ε : ℝ f : ι → α → β g : α → β inst✝² : SemilatticeSup ι inst✝¹ : Nonempty ι inst✝ : Countable ι hε : 0 < ε hf : ∀ (n : ι), StronglyMeasurable (f n) hg : StronglyMeasurable g hsm : MeasurableSet s hs : ↑↑μ s ≠ ⊤ hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x)) δ : ℝ hδ : δ > 0 N : ℕ hN : 1 / (↑N + 1) < δ n : ι hn : n ≥ notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N x : α hxs : x ∈ s hx : ∀ (k : ι), notConvergentSeqLTIndex (_ : 0 < ε / 2) hf hg hsm hs hfg N ≤ k → dist (f k x) (g x) ≤ 1 / (↑N + 1) ⊢ dist (f n x) (g x) < δ ** exact lt_of_le_of_lt (hx n hn) hN ** Qed
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MeasurableSpace.comap_const ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace β b : β ⊢ MeasurableSpace.comap (fun _a => b) m ≤ ⊥ ** rintro _ ⟨s, -, rfl⟩ ** case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace β b : β s : Set β ⊢ MeasurableSet ((fun _a => b) ⁻¹' s) ** by_cases b ∈ s <;> simp [*] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace β b : β s : Set β h : ¬b ∈ s ⊢ MeasurableSet ∅ ** exact measurableSet_empty _ ** Qed
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measurable_const' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f✝ g : α → β inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : β → α hf : ∀ (x y : β), f x = f y ⊢ Measurable f ** nontriviality β ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f✝ g : α → β inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : β → α hf : ∀ (x y : β), f x = f y ✝ : Nontrivial β ⊢ Measurable f ** inhabit β ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f✝ g : α → β inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : β → α hf : ∀ (x y : β), f x = f y ✝ : Nontrivial β inhabited_h : Inhabited β ⊢ Measurable f ** convert @measurable_const α β _ _ (f default) using 2 ** case h.e'_5.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f✝ g : α → β inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : β → α hf : ∀ (x y : β), f x = f y ✝ : Nontrivial β inhabited_h : Inhabited β x✝ : β ⊢ f x✝ = f default ** apply hf ** Qed
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Measurable.piecewise ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β x✝ : DecidablePred fun x => x ∈ s hs : MeasurableSet s hf : Measurable f hg : Measurable g ⊢ Measurable (piecewise s f g) ** intro t ht ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β x✝ : DecidablePred fun x => x ∈ s hs : MeasurableSet s hf : Measurable f hg : Measurable g t : Set β ht : MeasurableSet t ⊢ MeasurableSet (piecewise s f g ⁻¹' t) ** rw [piecewise_preimage] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β x✝ : DecidablePred fun x => x ∈ s hs : MeasurableSet s hf : Measurable f hg : Measurable g t : Set β ht : MeasurableSet t ⊢ MeasurableSet (Set.ite s (f ⁻¹' t) (g ⁻¹' t)) ** exact hs.ite (hf ht) (hg ht) ** Qed
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Measurable.measurable_of_countable_ne ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} ⊢ Measurable g ** intro t ht ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t ⊢ MeasurableSet (g ⁻¹' t) ** have : g ⁻¹' t = g ⁻¹' t ∩ { x | f x = g x }ᶜ ∪ g ⁻¹' t ∩ { x | f x = g x } := by
simp [← inter_union_distrib_left] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} ⊢ MeasurableSet (g ⁻¹' t) ** rw [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} ⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}) ** refine (h.mono (inter_subset_right _ _)).measurableSet.union ?_ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} ⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x}) ** have : g ⁻¹' t ∩ { x : α | f x = g x } = f ⁻¹' t ∩ { x : α | f x = g x } := by
ext x
simp (config := { contextual := true }) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t this✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} this : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x} ⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x}) ** rw [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t this✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} this : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x} ⊢ MeasurableSet (f ⁻¹' t ∩ {x | f x = g x}) ** exact (hf ht).inter h.measurableSet.of_compl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t ⊢ g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} ** simp [← inter_union_distrib_left] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} ⊢ g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x} ** ext x ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α f g : α → β m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSingletonClass α hf : Measurable f h : Set.Countable {x | f x ≠ g x} t : Set β ht : MeasurableSet t this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x} x : α ⊢ x ∈ g ⁻¹' t ∩ {x | f x = g x} ↔ x ∈ f ⁻¹' t ∩ {x | f x = g x} ** simp (config := { contextual := true }) ** Qed
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measurable_to_countable ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝² : MeasurableSpace α inst✝¹ : Countable α inst✝ : MeasurableSpace β f : β → α h : ∀ (y : β), MeasurableSet (f ⁻¹' {f y}) s : Set α x✝ : MeasurableSet s ⊢ MeasurableSet (f ⁻¹' s) ** rw [← biUnion_preimage_singleton] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝² : MeasurableSpace α inst✝¹ : Countable α inst✝ : MeasurableSpace β f : β → α h : ∀ (y : β), MeasurableSet (f ⁻¹' {f y}) s : Set α x✝ : MeasurableSet s ⊢ MeasurableSet (⋃ y ∈ s, f ⁻¹' {y}) ** refine' MeasurableSet.iUnion fun y => MeasurableSet.iUnion fun hy => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝² : MeasurableSpace α inst✝¹ : Countable α inst✝ : MeasurableSpace β f : β → α h : ∀ (y : β), MeasurableSet (f ⁻¹' {f y}) s : Set α x✝ : MeasurableSet s y : α hy : y ∈ s ⊢ MeasurableSet (f ⁻¹' {y}) ** by_cases hyf : y ∈ range f ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝² : MeasurableSpace α inst✝¹ : Countable α inst✝ : MeasurableSpace β f : β → α h : ∀ (y : β), MeasurableSet (f ⁻¹' {f y}) s : Set α x✝ : MeasurableSet s y : α hy : y ∈ s hyf : y ∈ range f ⊢ MeasurableSet (f ⁻¹' {y}) ** rcases hyf with ⟨y, rfl⟩ ** case pos.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝² : MeasurableSpace α inst✝¹ : Countable α inst✝ : MeasurableSpace β f : β → α h : ∀ (y : β), MeasurableSet (f ⁻¹' {f y}) s : Set α x✝ : MeasurableSet s y : β hy : f y ∈ s ⊢ MeasurableSet (f ⁻¹' {f y}) ** apply h ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝² : MeasurableSpace α inst✝¹ : Countable α inst✝ : MeasurableSpace β f : β → α h : ∀ (y : β), MeasurableSet (f ⁻¹' {f y}) s : Set α x✝ : MeasurableSet s y : α hy : y ∈ s hyf : ¬y ∈ range f ⊢ MeasurableSet (f ⁻¹' {y}) ** simp only [preimage_singleton_eq_empty.2 hyf, MeasurableSet.empty] ** Qed
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measurable_find ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝¹ : MeasurableSpace α p : α → ℕ → Prop inst✝ : (x : α) → DecidablePred (p x) hp : ∀ (x : α), ∃ N, p x N hm : ∀ (k : ℕ), MeasurableSet {x | p x k} ⊢ Measurable fun x => Nat.find (_ : ∃ N, p x N) ** refine' measurable_to_nat fun x => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝¹ : MeasurableSpace α p : α → ℕ → Prop inst✝ : (x : α) → DecidablePred (p x) hp : ∀ (x : α), ∃ N, p x N hm : ∀ (k : ℕ), MeasurableSet {x | p x k} x : α ⊢ MeasurableSet ((fun x => Nat.find (_ : ∃ N, p x N)) ⁻¹' {Nat.find (_ : ∃ N, p x N)}) ** rw [preimage_find_eq_disjointed (fun k => {x | p x k})] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝¹ : MeasurableSpace α p : α → ℕ → Prop inst✝ : (x : α) → DecidablePred (p x) hp : ∀ (x : α), ∃ N, p x N hm : ∀ (k : ℕ), MeasurableSet {x | p x k} x : α ⊢ MeasurableSet (disjointed (fun k => {x | p x k}) (Nat.find (_ : ∃ N, p x N))) ** exact MeasurableSet.disjointed hm _ ** Qed
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MeasurableSet.subtype_image ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α m : MeasurableSpace α mβ : MeasurableSpace β s : Set α t : Set ↑s hs : MeasurableSet s ⊢ MeasurableSet t → MeasurableSet (Subtype.val '' t) ** rintro ⟨u, hu, rfl⟩ ** case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s : Set α hs : MeasurableSet s u : Set α hu : MeasurableSet u ⊢ MeasurableSet (Subtype.val '' (Subtype.val ⁻¹' u)) ** rw [Subtype.image_preimage_coe] ** case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s : Set α hs : MeasurableSet s u : Set α hu : MeasurableSet u ⊢ MeasurableSet (u ∩ s) ** exact hu.inter hs ** Qed
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MeasurableSet.image_inclusion' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s t : Set α h : s ⊆ t u : Set ↑s hs : MeasurableSet (Subtype.val ⁻¹' s) hu : MeasurableSet u ⊢ MeasurableSet (inclusion h '' u) ** rcases hu with ⟨u, hu, rfl⟩ ** case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s t : Set α h : s ⊆ t hs : MeasurableSet (Subtype.val ⁻¹' s) u : Set α hu : MeasurableSet u ⊢ MeasurableSet (inclusion h '' (Subtype.val ⁻¹' u)) ** convert (measurable_subtype_coe hu).inter hs ** case h.e'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s t : Set α h : s ⊆ t hs : MeasurableSet (Subtype.val ⁻¹' s) u : Set α hu : MeasurableSet u ⊢ inclusion h '' (Subtype.val ⁻¹' u) = Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s ** ext ⟨x, hx⟩ ** case h.e'_3.h.mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s t : Set α h : s ⊆ t hs : MeasurableSet (Subtype.val ⁻¹' s) u : Set α hu : MeasurableSet u x : α hx : x ∈ t ⊢ { val := x, property := hx } ∈ inclusion h '' (Subtype.val ⁻¹' u) ↔ { val := x, property := hx } ∈ Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s ** simpa [@and_comm _ (_ = x)] using and_comm ** Qed
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MeasurableSet.of_union_cover ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s t u : Set α hs : MeasurableSet s ht : MeasurableSet t h : univ ⊆ s ∪ t hsu : MeasurableSet (Subtype.val ⁻¹' u) htu : MeasurableSet (Subtype.val ⁻¹' u) ⊢ MeasurableSet u ** convert (hs.subtype_image hsu).union (ht.subtype_image htu) ** case h.e'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u✝ : Set α m : MeasurableSpace α mβ : MeasurableSpace β s t u : Set α hs : MeasurableSet s ht : MeasurableSet t h : univ ⊆ s ∪ t hsu : MeasurableSet (Subtype.val ⁻¹' u) htu : MeasurableSet (Subtype.val ⁻¹' u) ⊢ u = Subtype.val '' (Subtype.val ⁻¹' u) ∪ Subtype.val '' (Subtype.val ⁻¹' u) ** simp [image_preimage_eq_inter_range, ← inter_distrib_left, univ_subset_iff.1 h] ** Qed
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Measurable.dite ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : (x : α) → Decidable (x ∈ s) f : ↑s → β hf : Measurable f g : ↑sᶜ → β hg : Measurable g hs : MeasurableSet s ⊢ Measurable (restrict s fun x => if hx : x ∈ s then f { val := x, property := hx } else g { val := x, property := hx }) ** simpa ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : (x : α) → Decidable (x ∈ s) f : ↑s → β hf : Measurable f g : ↑sᶜ → β hg : Measurable g hs : MeasurableSet s ⊢ Measurable (restrict sᶜ fun x => if hx : x ∈ s then f { val := x, property := hx } else g { val := x, property := hx }) ** simpa ** Qed
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Measurable.prod ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ f : α → β × γ hf₁ : Measurable fun a => (f a).1 hf₂ : Measurable fun a => (f a).2 ⊢ MeasurableSpace.comap Prod.fst mβ ≤ MeasurableSpace.map f m ** rw [MeasurableSpace.comap_le_iff_le_map, MeasurableSpace.map_comp] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ f : α → β × γ hf₁ : Measurable fun a => (f a).1 hf₂ : Measurable fun a => (f a).2 ⊢ mβ ≤ MeasurableSpace.map (Prod.fst ∘ f) m ** exact hf₁ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ f : α → β × γ hf₁ : Measurable fun a => (f a).1 hf₂ : Measurable fun a => (f a).2 ⊢ MeasurableSpace.comap Prod.snd mγ ≤ MeasurableSpace.map f m ** rw [MeasurableSpace.comap_le_iff_le_map, MeasurableSpace.map_comp] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ f : α → β × γ hf₁ : Measurable fun a => (f a).1 hf₂ : Measurable fun a => (f a).2 ⊢ mγ ≤ MeasurableSpace.map (Prod.snd ∘ f) m ** exact hf₂ ** Qed
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measurableSet_prod_of_nonempty ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ s : Set α t : Set β h : Set.Nonempty (s ×ˢ t) ⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ** rcases h with ⟨⟨x, y⟩, hx, hy⟩ ** case intro.mk.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ s : Set α t : Set β x : α y : β hx : (x, y).1 ∈ s hy : (x, y).2 ∈ t ⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ** refine' ⟨fun hst => _, fun h => h.1.prod h.2⟩ ** case intro.mk.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ s : Set α t : Set β x : α y : β hx : (x, y).1 ∈ s hy : (x, y).2 ∈ t hst : MeasurableSet (s ×ˢ t) ⊢ MeasurableSet s ∧ MeasurableSet t ** have : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t) := measurable_prod_mk_right hst ** case intro.mk.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ s : Set α t : Set β x : α y : β hx : (x, y).1 ∈ s hy : (x, y).2 ∈ t hst : MeasurableSet (s ×ˢ t) this : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t) ⊢ MeasurableSet s ∧ MeasurableSet t ** have : MeasurableSet (Prod.mk x ⁻¹' s ×ˢ t) := measurable_prod_mk_left hst ** case intro.mk.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ s : Set α t : Set β x : α y : β hx : (x, y).1 ∈ s hy : (x, y).2 ∈ t hst : MeasurableSet (s ×ˢ t) this✝ : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t) this : MeasurableSet (Prod.mk x ⁻¹' s ×ˢ t) ⊢ MeasurableSet s ∧ MeasurableSet t ** simp_all ** Qed
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measurable_from_prod_countable ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝¹ : Countable β inst✝ : MeasurableSingletonClass β x✝ : MeasurableSpace γ f : α × β → γ hf : ∀ (y : β), Measurable fun x => f (x, y) s : Set γ hs : MeasurableSet s ⊢ MeasurableSet (f ⁻¹' s) ** have : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ ({y} : Set β) := by
ext1 ⟨x, y⟩
simp [and_assoc, and_left_comm] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝¹ : Countable β inst✝ : MeasurableSingletonClass β x✝ : MeasurableSpace γ f : α × β → γ hf : ∀ (y : β), Measurable fun x => f (x, y) s : Set γ hs : MeasurableSet s this : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ {y} ⊢ MeasurableSet (f ⁻¹' s) ** rw [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝¹ : Countable β inst✝ : MeasurableSingletonClass β x✝ : MeasurableSpace γ f : α × β → γ hf : ∀ (y : β), Measurable fun x => f (x, y) s : Set γ hs : MeasurableSet s this : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ {y} ⊢ MeasurableSet (⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ {y}) ** exact .iUnion fun y => (hf y hs).prod (.singleton y) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝¹ : Countable β inst✝ : MeasurableSingletonClass β x✝ : MeasurableSpace γ f : α × β → γ hf : ∀ (y : β), Measurable fun x => f (x, y) s : Set γ hs : MeasurableSet s ⊢ f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ {y} ** ext1 ⟨x, y⟩ ** case h.mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝¹ : Countable β inst✝ : MeasurableSingletonClass β x✝ : MeasurableSpace γ f : α × β → γ hf : ∀ (y : β), Measurable fun x => f (x, y) s : Set γ hs : MeasurableSet s x : α y : β ⊢ (x, y) ∈ f ⁻¹' s ↔ (x, y) ∈ ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ {y} ** simp [and_assoc, and_left_comm] ** Qed
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exists_measurable_piecewise_nat ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t✝ u : Set α m✝ : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ m : MeasurableSpace α t : ℕ → Set β t_meas : ∀ (n : ℕ), MeasurableSet (t n) t_disj : Pairwise (Disjoint on t) g : ℕ → β → α hg : ∀ (n : ℕ), Measurable (g n) i j : ℕ h : (Disjoint on t) i j ⊢ EqOn (g i) (g j) (t i ∩ t j) ** simp only [h.inter_eq, eqOn_empty] ** Qed
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measurable_update' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ a : δ inst✝ : DecidableEq δ ⊢ Measurable fun p => update p.1 a p.2 ** rw [measurable_pi_iff] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ a : δ inst✝ : DecidableEq δ ⊢ ∀ (a_1 : δ), Measurable fun x => update x.1 a x.2 a_1 ** intro j ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ a : δ inst✝ : DecidableEq δ j : δ ⊢ Measurable fun x => update x.1 a x.2 j ** dsimp [update] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ a : δ inst✝ : DecidableEq δ j : δ ⊢ Measurable fun x => if h : j = a then (_ : a = j) ▸ x.2 else x.1 j ** split_ifs with h ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ a : δ inst✝ : DecidableEq δ j : δ h : j = a ⊢ Measurable fun x => (_ : a = j) ▸ x.2 ** subst h ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ inst✝ : DecidableEq δ j : δ ⊢ Measurable fun x => (_ : j = j) ▸ x.2 ** dsimp ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ inst✝ : DecidableEq δ j : δ ⊢ Measurable fun x => x.2 ** exact measurable_snd ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ a : δ inst✝ : DecidableEq δ j : δ h : ¬j = a ⊢ Measurable fun x => x.1 j ** exact measurable_pi_iff.1 measurable_fst _ ** Qed
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measurable_eq_mp ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝² : MeasurableSpace α inst✝¹ : (a : δ) → MeasurableSpace (π a) inst✝ : MeasurableSpace γ i i' : δ h : i = i' ⊢ Measurable (Eq.mp (_ : π i = π i')) ** cases h ** case refl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝² : MeasurableSpace α inst✝¹ : (a : δ) → MeasurableSpace (π a) inst✝ : MeasurableSpace γ i : δ ⊢ Measurable (Eq.mp (_ : π i = π i)) ** exact measurable_id ** Qed
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MeasurableSet.pi ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α π : δ → Type u_6 inst✝² : MeasurableSpace α inst✝¹ : (a : δ) → MeasurableSpace (π a) inst✝ : MeasurableSpace γ s : Set δ t : (i : δ) → Set (π i) hs : Set.Countable s ht : ∀ (i : δ), i ∈ s → MeasurableSet (t i) ⊢ MeasurableSet (Set.pi s t) ** rw [pi_def] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α π : δ → Type u_6 inst✝² : MeasurableSpace α inst✝¹ : (a : δ) → MeasurableSpace (π a) inst✝ : MeasurableSpace γ s : Set δ t : (i : δ) → Set (π i) hs : Set.Countable s ht : ∀ (i : δ), i ∈ s → MeasurableSet (t i) ⊢ MeasurableSet (⋂ a ∈ s, eval a ⁻¹' t a) ** exact MeasurableSet.biInter hs fun i hi => measurable_pi_apply _ (ht i hi) ** Qed
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measurable_piEquivPiSubtypeProd_symm ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p ⊢ Measurable ↑(piEquivPiSubtypeProd p π).symm ** refine' measurable_pi_iff.2 fun j => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p j : δ ⊢ Measurable fun x => ↑(piEquivPiSubtypeProd p π).symm x j ** by_cases hj : p j ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p j : δ hj : p j ⊢ Measurable fun x => ↑(piEquivPiSubtypeProd p π).symm x j ** simp only [hj, dif_pos, Equiv.piEquivPiSubtypeProd_symm_apply] ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p j : δ hj : p j ⊢ Measurable fun x => x.1 { val := j, property := (_ : p j) } ** have : Measurable fun (f : ∀ i : { x // p x }, π i.1) => f ⟨j, hj⟩ :=
measurable_pi_apply (π := fun i : {x // p x} => π i.1) ⟨j, hj⟩ ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p j : δ hj : p j this : Measurable fun f => f { val := j, property := hj } ⊢ Measurable fun x => x.1 { val := j, property := (_ : p j) } ** exact Measurable.comp this measurable_fst ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p j : δ hj : ¬p j ⊢ Measurable fun x => ↑(piEquivPiSubtypeProd p π).symm x j ** simp only [hj, Equiv.piEquivPiSubtypeProd_symm_apply, dif_neg, not_false_iff] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p j : δ hj : ¬p j ⊢ Measurable fun x => x.2 { val := j, property := (_ : ¬p j) } ** have : Measurable fun (f : ∀ i : { x // ¬p x }, π i.1) => f ⟨j, hj⟩ :=
measurable_pi_apply (π := fun i : {x // ¬p x} => π i.1) ⟨j, hj⟩ ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝³ : MeasurableSpace α inst✝² : (a : δ) → MeasurableSpace (π a) inst✝¹ : MeasurableSpace γ p : δ → Prop inst✝ : DecidablePred p j : δ hj : ¬p j this : Measurable fun f => f { val := j, property := hj } ⊢ Measurable fun x => x.2 { val := j, property := (_ : ¬p j) } ** exact Measurable.comp this measurable_snd ** Qed
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measurable_tProd_mk ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝ : (x : δ) → MeasurableSpace (π x) l : List δ ⊢ Measurable (TProd.mk l) ** induction' l with i l ih ** case nil α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝ : (x : δ) → MeasurableSpace (π x) ⊢ Measurable (TProd.mk []) ** exact measurable_const ** case cons α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝ : (x : δ) → MeasurableSpace (π x) i : δ l : List δ ih : Measurable (TProd.mk l) ⊢ Measurable (TProd.mk (i :: l)) ** exact (measurable_pi_apply i).prod_mk ih ** Qed
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measurable_tProd_elim ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝¹ : (x : δ) → MeasurableSpace (π x) inst✝ : DecidableEq δ i : δ is : List δ j : δ hj : j ∈ i :: is ⊢ Measurable fun v => TProd.elim v hj ** by_cases hji : j = i ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝¹ : (x : δ) → MeasurableSpace (π x) inst✝ : DecidableEq δ i : δ is : List δ j : δ hj : j ∈ i :: is hji : j = i ⊢ Measurable fun v => TProd.elim v hj ** subst hji ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝¹ : (x : δ) → MeasurableSpace (π x) inst✝ : DecidableEq δ is : List δ j : δ hj : j ∈ j :: is ⊢ Measurable fun v => TProd.elim v hj ** simpa using measurable_fst ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝¹ : (x : δ) → MeasurableSpace (π x) inst✝ : DecidableEq δ i : δ is : List δ j : δ hj : j ∈ i :: is hji : ¬j = i ⊢ Measurable fun v => TProd.elim v hj ** simp only [TProd.elim_of_ne _ hji] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝¹ : (x : δ) → MeasurableSpace (π x) inst✝ : DecidableEq δ i : δ is : List δ j : δ hj : j ∈ i :: is hji : ¬j = i ⊢ Measurable fun v => TProd.elim v.2 (_ : j ∈ is) ** rw [mem_cons] at hj ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α π : δ → Type u_6 inst✝¹ : (x : δ) → MeasurableSpace (π x) inst✝ : DecidableEq δ i : δ is : List δ j : δ hj✝ : j ∈ i :: is hj : j = i ∨ j ∈ is hji : ¬j = i ⊢ Measurable fun v => TProd.elim v.2 (_ : j ∈ is) ** exact (measurable_tProd_elim (hj.resolve_left hji)).comp measurable_snd ** Qed
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MeasurableSet.tProd ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α π : δ → Type u_6 inst✝ : (x : δ) → MeasurableSpace (π x) l : List δ s : (i : δ) → Set (π i) hs : ∀ (i : δ), MeasurableSet (s i) ⊢ MeasurableSet (Set.tprod l s) ** induction' l with i l ih ** case nil α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α π : δ → Type u_6 inst✝ : (x : δ) → MeasurableSpace (π x) s : (i : δ) → Set (π i) hs : ∀ (i : δ), MeasurableSet (s i) ⊢ MeasurableSet (Set.tprod [] s) case cons α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α π : δ → Type u_6 inst✝ : (x : δ) → MeasurableSpace (π x) s : (i : δ) → Set (π i) hs : ∀ (i : δ), MeasurableSet (s i) i : δ l : List δ ih : MeasurableSet (Set.tprod l s) ⊢ MeasurableSet (Set.tprod (i :: l) s) ** exact MeasurableSet.univ ** case cons α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α π : δ → Type u_6 inst✝ : (x : δ) → MeasurableSpace (π x) s : (i : δ) → Set (π i) hs : ∀ (i : δ), MeasurableSet (s i) i : δ l : List δ ih : MeasurableSet (Set.tprod l s) ⊢ MeasurableSet (Set.tprod (i :: l) s) ** exact (hs i).prod ih ** Qed
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measurableSet_inl_image ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β s : Set α ⊢ MeasurableSet (Sum.inl '' s) ↔ MeasurableSet s ** simp [measurableSet_sum_iff, Sum.inl_injective.preimage_image] ** Qed
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measurableSet_inr_image ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β s : Set β ⊢ MeasurableSet (Sum.inr '' s) ↔ MeasurableSet s ** simp [measurableSet_sum_iff, Sum.inr_injective.preimage_image] ** Qed
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measurableSet_range_inl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSpace α ⊢ MeasurableSet (range Sum.inl) ** rw [← image_univ] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSpace α ⊢ MeasurableSet (Sum.inl '' univ) ** exact MeasurableSet.univ.inl_image ** Qed
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measurableSet_range_inr ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSpace α ⊢ MeasurableSet (range Sum.inr) ** rw [← image_univ] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace α mβ : MeasurableSpace β inst✝ : MeasurableSpace α ⊢ MeasurableSet (Sum.inr '' univ) ** exact MeasurableSet.univ.inr_image ** Qed
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MeasurableEmbedding.measurableSet_image ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hf : MeasurableEmbedding f s : Set α h : MeasurableSet (f '' s) ⊢ MeasurableSet s ** simpa only [hf.injective.preimage_image] using hf.measurable h ** Qed
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MeasurableEmbedding.id ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ s : Set α hs : MeasurableSet s ⊢ MeasurableSet (_root_.id '' s) ** rwa [image_id] ** Qed
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MeasurableEmbedding.comp ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hg : MeasurableEmbedding g hf : MeasurableEmbedding f s : Set α hs : MeasurableSet s ⊢ MeasurableSet (g ∘ f '' s) ** rwa [image_comp, hg.measurableSet_image, hf.measurableSet_image] ** Qed
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MeasurableEmbedding.measurableSet_range ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hf : MeasurableEmbedding f ⊢ MeasurableSet (range f) ** rw [← image_univ] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hf : MeasurableEmbedding f ⊢ MeasurableSet (f '' univ) ** exact hf.measurableSet_image' MeasurableSet.univ ** Qed
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MeasurableEmbedding.measurableSet_preimage ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hf : MeasurableEmbedding f s : Set β ⊢ MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f) ** rw [← image_preimage_eq_inter_range, hf.measurableSet_image] ** Qed
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MeasurableEmbedding.measurable_comp_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hg : MeasurableEmbedding g ⊢ Measurable (g ∘ f) ↔ Measurable f ** refine' ⟨fun H => _, hg.measurable.comp⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hg : MeasurableEmbedding g H : Measurable (g ∘ f) ⊢ Measurable f ** suffices Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) by
rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hg : MeasurableEmbedding g H : Measurable (g ∘ f) ⊢ Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) ** exact hg.measurable_rangeSplitting.comp H.subtype_mk ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α mα : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → γ hg : MeasurableEmbedding g H : Measurable (g ∘ f) this : Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) ⊢ Measurable f ** rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this ** Qed
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MeasurableEquiv.toEquiv_injective ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ ⊢ Injective toEquiv ** rintro ⟨e₁, _, _⟩ ⟨e₂, _, _⟩ (rfl : e₁ = e₂) ** case mk.mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ e₁ : α ≃ β measurable_toFun✝¹ : Measurable ↑e₁ measurable_invFun✝¹ : Measurable ↑e₁.symm measurable_toFun✝ : Measurable ↑e₁ measurable_invFun✝ : Measurable ↑e₁.symm ⊢ { toEquiv := e₁, measurable_toFun := measurable_toFun✝¹, measurable_invFun := measurable_invFun✝¹ } = { toEquiv := e₁, measurable_toFun := measurable_toFun✝, measurable_invFun := measurable_invFun✝ } ** rfl ** Qed
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MeasurableEquiv.coe_piCongrLeft ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝⁵ : MeasurableSpace α inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ π : δ' → Type u_6 π' : δ' → Type u_7 inst✝¹ : (x : δ') → MeasurableSpace (π x) inst✝ : (x : δ') → MeasurableSpace (π' x) f : δ ≃ δ' ⊢ ↑(piCongrLeft π f) = ↑(Equiv.piCongrLeft π f) ** rfl ** Qed
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MeasurableEquiv.coe_sumPiEquivProdPi ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α✝ inst✝⁶ : MeasurableSpace α✝ inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace γ inst✝³ : MeasurableSpace δ π : δ' → Type u_6 π' : δ' → Type u_7 inst✝² : (x : δ') → MeasurableSpace (π x) inst✝¹ : (x : δ') → MeasurableSpace (π' x) α : δ ⊕ δ' → Type u_8 inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i) ⊢ ↑(sumPiEquivProdPi α) = ↑(Equiv.sumPiEquivProdPi α) ** rfl ** Qed
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MeasurableEquiv.coe_sumPiEquivProdPi_symm ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α✝ inst✝⁶ : MeasurableSpace α✝ inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace γ inst✝³ : MeasurableSpace δ π : δ' → Type u_6 π' : δ' → Type u_7 inst✝² : (x : δ') → MeasurableSpace (π x) inst✝¹ : (x : δ') → MeasurableSpace (π' x) α : δ ⊕ δ' → Type u_8 inst✝ : (i : δ ⊕ δ') → MeasurableSpace (α i) ⊢ ↑(symm (sumPiEquivProdPi α)) = ↑(Equiv.sumPiEquivProdPi α).symm ** rfl ** Qed
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MeasurableEmbedding.iff_comap_eq ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → α hf : Injective f ∧ MeasurableSpace.comap f inst✝¹ = inst✝² ∧ MeasurableSet (range f) ⊢ Measurable f ** exact comap_measurable f ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → α hf : Injective f ∧ MeasurableSpace.comap f inst✝¹ = inst✝² ∧ MeasurableSet (range f) ⊢ ∀ ⦃s : Set α⦄, MeasurableSet s → MeasurableSet (f '' s) ** rintro _ ⟨s, hs, rfl⟩ ** case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g : β → α hf : Injective f ∧ MeasurableSpace.comap f inst✝¹ = inst✝² ∧ MeasurableSet (range f) s : Set β hs : MeasurableSet s ⊢ MeasurableSet (f '' (f ⁻¹' s)) ** simpa only [image_preimage_eq_inter_range] using hs.inter hf.2.2 ** Qed
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MeasurableEmbedding.of_measurable_inverse_on_range ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g✝ : β → α g : ↑(range f) → α hf₁ : Measurable f hf₂ : MeasurableSet (range f) hg : Measurable g H : LeftInverse g (rangeFactorization f) ⊢ MeasurableEmbedding f ** set e : α ≃ᵐ range f :=
⟨⟨rangeFactorization f, g, H, H.rightInverse_of_surjective surjective_onto_range⟩,
hf₁.subtype_mk, hg⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β inst✝ : MeasurableSpace γ f : α → β g✝ : β → α g : ↑(range f) → α hf₁ : Measurable f hf₂ : MeasurableSet (range f) hg : Measurable g H : LeftInverse g (rangeFactorization f) e : α ≃ᵐ ↑(range f) := { toEquiv := { toFun := rangeFactorization f, invFun := g, left_inv := H, right_inv := (_ : Function.RightInverse g (rangeFactorization f)) }, measurable_toFun := (_ : Measurable fun x => { val := f x, property := (_ : f x ∈ range f) }), measurable_invFun := hg } ⊢ MeasurableEmbedding f ** exact (MeasurableEmbedding.subtype_coe hf₂).comp e.measurableEmbedding ** Qed
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MeasurableSpace.CountablyGenerated.comap ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α m : MeasurableSpace β h : CountablyGenerated β f : α → β ⊢ CountablyGenerated α ** rcases h with ⟨⟨b, hbc, rfl⟩⟩ ** case mk.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f : α → β b : Set (Set β) hbc : Set.Countable b ⊢ CountablyGenerated α ** letI := generateFrom (preimage f '' b) ** case mk.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s t u : Set α f : α → β b : Set (Set β) hbc : Set.Countable b this : MeasurableSpace α := generateFrom (preimage f '' b) ⊢ CountablyGenerated α ** exact ⟨_, hbc.image _, rfl⟩ ** Qed
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Filter.principal_isMeasurablyGenerated_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝ : MeasurableSpace α s : Set α ⊢ IsMeasurablyGenerated (𝓟 s) ↔ MeasurableSet s ** refine' ⟨_, fun hs => ⟨fun t ht => ⟨s, mem_principal_self s, hs, ht⟩⟩⟩ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝ : MeasurableSpace α s : Set α ⊢ IsMeasurablyGenerated (𝓟 s) → MeasurableSet s ** rintro ⟨hs⟩ ** case mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝ : MeasurableSpace α s : Set α hs : ∀ ⦃s_1 : Set α⦄, s_1 ∈ 𝓟 s → ∃ t, t ∈ 𝓟 s ∧ MeasurableSet t ∧ t ⊆ s_1 ⊢ MeasurableSet s ** rcases hs (mem_principal_self s) with ⟨t, ht, htm, hts⟩ ** case mk.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α inst✝ : MeasurableSpace α s : Set α hs : ∀ ⦃s_1 : Set α⦄, s_1 ∈ 𝓟 s → ∃ t, t ∈ 𝓟 s ∧ MeasurableSet t ∧ t ⊆ s_1 t : Set α ht : t ∈ 𝓟 s htm : MeasurableSet t hts : t ⊆ s ⊢ MeasurableSet s ** have : t = s := hts.antisymm ht ** case mk.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t✝ u : Set α inst✝ : MeasurableSpace α s : Set α hs : ∀ ⦃s_1 : Set α⦄, s_1 ∈ 𝓟 s → ∃ t, t ∈ 𝓟 s ∧ MeasurableSet t ∧ t ⊆ s_1 t : Set α ht : t ∈ 𝓟 s htm : MeasurableSet t hts : t ⊆ s this : t = s ⊢ MeasurableSet s ** rwa [← this] ** Qed
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MeasurableSet.measurableSet_blimsup ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝ : MeasurableSpace α s : ℕ → Set α p : ℕ → Prop h : ∀ (n : ℕ), p n → MeasurableSet (s n) ⊢ MeasurableSet (blimsup s atTop p) ** simp only [blimsup_eq_iInf_biSup_of_nat, iSup_eq_iUnion, iInf_eq_iInter] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 δ' : Type u_5 ι : Sort uι s✝ t u : Set α inst✝ : MeasurableSpace α s : ℕ → Set α p : ℕ → Prop h : ∀ (n : ℕ), p n → MeasurableSet (s n) ⊢ MeasurableSet (⋂ i, ⋃ j, ⋃ (_ : p j ∧ i ≤ j), s j) ** exact .iInter fun _ => .iUnion fun m => .iUnion fun hm => h m hm.1 ** Qed
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MeasureTheory.StronglyMeasurable.integral_prod_right' ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : SigmaFinite ν f : α × β → E hf : StronglyMeasurable f ⊢ StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν ** rw [← uncurry_curry f] at hf ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : SigmaFinite ν f : α × β → E hf : StronglyMeasurable (uncurry (curry f)) ⊢ StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν ** exact hf.integral_prod_right ** Qed
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MeasureTheory.Measure.integrable_measure_prod_mk_left ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ Integrable fun x => ENNReal.toReal (↑↑ν (Prod.mk x ⁻¹' s)) ** refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ HasFiniteIntegral fun x => ENNReal.toReal (↑↑ν (Prod.mk x ⁻¹' s)) ** simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s))) ∂μ < ⊤ ** convert h2s.lt_top using 1 ** case h.e'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s))) ∂μ = ↑↑(Measure.prod μ ν) s ** rw [prod_apply hs] ** case h.e'_3 α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s))) ∂μ = ∫⁻ (x : α), ↑↑ν (Prod.mk x ⁻¹' s) ∂μ ** apply lintegral_congr_ae ** case h.e'_3.h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s)))) =ᶠ[ae μ] fun a => ↑↑ν (Prod.mk a ⁻¹' s) ** refine' (ae_measure_lt_top hs h2s).mp _ ** case h.e'_3.h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, ↑↑ν (Prod.mk x ⁻¹' s) < ⊤ → (fun a => ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s)))) x = (fun a => ↑↑ν (Prod.mk a ⁻¹' s)) x ** apply eventually_of_forall ** case h.e'_3.h.hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ ⊢ ∀ (x : α), ↑↑ν (Prod.mk x ⁻¹' s) < ⊤ → (fun a => ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s)))) x = (fun a => ↑↑ν (Prod.mk a ⁻¹' s)) x ** intro x hx ** case h.e'_3.h.hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ x : α hx : ↑↑ν (Prod.mk x ⁻¹' s) < ⊤ ⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s)))) x = (fun a => ↑↑ν (Prod.mk a ⁻¹' s)) x ** rw [lt_top_iff_ne_top] at hx ** case h.e'_3.h.hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν s : Set (α × β) hs : MeasurableSet s h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤ x : α hx : ↑↑ν (Prod.mk x ⁻¹' s) ≠ ⊤ ⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (↑↑ν (Prod.mk a ⁻¹' s)))) x = (fun a => ↑↑ν (Prod.mk a ⁻¹' s)) x ** simp [ofReal_toReal, hx] ** Qed
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MeasureTheory.AEStronglyMeasurable.prod_swap ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ✝ μ μ' : Measure α ν ν' : Measure β τ : Measure γ✝ inst✝³ : NormedAddCommGroup E γ : Type u_7 inst✝² : TopologicalSpace γ inst✝¹ : SigmaFinite μ inst✝ : SigmaFinite ν f : β × α → γ hf : AEStronglyMeasurable f (Measure.prod ν μ) ⊢ AEStronglyMeasurable (fun z => f (Prod.swap z)) (Measure.prod μ ν) ** rw [← prod_swap] at hf ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁸ : MeasurableSpace α inst✝⁷ : MeasurableSpace α' inst✝⁶ : MeasurableSpace β inst✝⁵ : MeasurableSpace β' inst✝⁴ : MeasurableSpace γ✝ μ μ' : Measure α ν ν' : Measure β τ : Measure γ✝ inst✝³ : NormedAddCommGroup E γ : Type u_7 inst✝² : TopologicalSpace γ inst✝¹ : SigmaFinite μ inst✝ : SigmaFinite ν f : β × α → γ hf : AEStronglyMeasurable f (map Prod.swap (Measure.prod μ ν)) ⊢ AEStronglyMeasurable (fun z => f (Prod.swap z)) (Measure.prod μ ν) ** exact hf.comp_measurable measurable_swap ** Qed
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MeasureTheory.AEStronglyMeasurable.integral_prod_right' ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : NormedSpace ℝ E f : α × β → E hf : AEStronglyMeasurable f (Measure.prod μ ν) ⊢ (fun x => ∫ (y : β), f (x, y) ∂ν) =ᶠ[ae μ] fun x => ∫ (y : β), AEStronglyMeasurable.mk f hf (x, y) ∂ν ** filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ hx using integral_congr_ae hx ** Qed
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MeasureTheory.AEStronglyMeasurable.prod_mk_left ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ✝ μ μ' : Measure α ν ν' : Measure β τ : Measure γ✝ inst✝² : NormedAddCommGroup E γ : Type u_7 inst✝¹ : SigmaFinite ν inst✝ : TopologicalSpace γ f : α × β → γ hf : AEStronglyMeasurable f (Measure.prod μ ν) ⊢ ∀ᵐ (x : α) ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν ** filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with x hx ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ✝ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ✝ μ μ' : Measure α ν ν' : Measure β τ : Measure γ✝ inst✝² : NormedAddCommGroup E γ : Type u_7 inst✝¹ : SigmaFinite ν inst✝ : TopologicalSpace γ f : α × β → γ hf : AEStronglyMeasurable f (Measure.prod μ ν) x : α hx : ∀ᵐ (y : β) ∂ν, f (x, y) = AEStronglyMeasurable.mk f hf (x, y) ⊢ AEStronglyMeasurable (fun y => f (x, y)) ν ** exact
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩ ** Qed
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MeasureTheory.hasFiniteIntegral_prod_iff ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f ⊢ HasFiniteIntegral f ↔ (∀ᵐ (x : α) ∂μ, HasFiniteIntegral fun y => f (x, y)) ∧ HasFiniteIntegral fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν ** simp only [HasFiniteIntegral] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f ⊢ ∫⁻ (a : α × β), ↑‖f a‖₊ ∂Measure.prod μ ν < ⊤ ↔ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤) ∧ ∫⁻ (a : α), ↑‖∫ (y : β), ‖f (a, y)‖ ∂ν‖₊ ∂μ < ⊤ ** rw [lintegral_prod_of_measurable _ h1f.ennnorm] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f ⊢ ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ↔ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤) ∧ ∫⁻ (a : α), ↑‖∫ (y : β), ‖f (a, y)‖ ∂ν‖₊ ∂μ < ⊤ ** have : ∀ x, ∀ᵐ y ∂ν, 0 ≤ ‖f (x, y)‖ := fun x => eventually_of_forall fun y => norm_nonneg _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ ⊢ ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ↔ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤) ∧ ∫⁻ (a : α), ↑‖∫ (y : β), ‖f (a, y)‖ ∂ν‖₊ ∂μ < ⊤ ** simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
ennnorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_coe_nnnorm] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ ⊢ ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ↔ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤) ∧ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν)) ∂μ < ⊤ ** have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
rw [← and_congr_right_iff, and_iff_right_of_imp h1] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) ⊢ ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ↔ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤) ∧ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν)) ∂μ < ⊤ ** rw [this] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ p q r : Prop h1 : r → p ⊢ (r ↔ p ∧ q) ↔ p → (r ↔ q) ** rw [← and_congr_right_iff, and_iff_right_of_imp h1] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) ⊢ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤) → (∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ↔ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν)) ∂μ < ⊤) ** intro h2f ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ↔ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν)) ∂μ < ⊤ ** rw [lintegral_congr_ae] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ (fun x => ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν) =ᶠ[ae μ] fun a => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν)) ** refine' h2f.mp _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ → (fun x => ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν) x = (fun a => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν))) x ** apply eventually_of_forall ** case hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ ∀ (x : α), ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ → (fun x => ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν) x = (fun a => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν))) x ** intro x hx ** case hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ x : α hx : ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ (fun x => ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν) x = (fun a => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν))) x ** dsimp only ** case hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ x : α hx : ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν = ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν)) ** rw [ofReal_toReal] ** case hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ x : α hx : ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν ≠ ⊤ ** rw [← lt_top_iff_ne_top] ** case hp α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ x : α hx : ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ⊢ ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ** exact hx ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) ⊢ ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ → ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ** intro h2f ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ⊢ ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤ ** refine' ae_lt_top _ h2f.ne ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁶ : MeasurableSpace α inst✝⁵ : MeasurableSpace α' inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace β' inst✝² : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝¹ : NormedAddCommGroup E inst✝ : SigmaFinite ν f : α × β → E h1f : StronglyMeasurable f this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ⊢ Measurable fun x => ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν ** exact h1f.ennnorm.lintegral_prod_right' ** Qed
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MeasureTheory.integrable_prod_iff' ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ f : α × β → E h1f : AEStronglyMeasurable f (Measure.prod μ ν) ⊢ Integrable f ↔ (∀ᵐ (y : β) ∂ν, Integrable fun x => f (x, y)) ∧ Integrable fun y => ∫ (x : α), ‖f (x, y)‖ ∂μ ** convert integrable_prod_iff h1f.prod_swap using 1 ** case h.e'_1.a α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝⁷ : MeasurableSpace α inst✝⁶ : MeasurableSpace α' inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace β' inst✝³ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝² : NormedAddCommGroup E inst✝¹ : SigmaFinite ν inst✝ : SigmaFinite μ f : α × β → E h1f : AEStronglyMeasurable f (Measure.prod μ ν) ⊢ Integrable f ↔ Integrable fun z => f (Prod.swap z) ** rw [funext fun _ => Function.comp_apply.symm, integrable_swap_iff] ** Qed
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MeasureTheory.integral_fn_integral_add ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → E' hf : Integrable f hg : Integrable g ⊢ ∫ (x : α), F (∫ (y : β), f (x, y) + g (x, y) ∂ν) ∂μ = ∫ (x : α), F (∫ (y : β), f (x, y) ∂ν + ∫ (y : β), g (x, y) ∂ν) ∂μ ** refine' integral_congr_ae _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → E' hf : Integrable f hg : Integrable g ⊢ (fun x => F (∫ (y : β), f (x, y) + g (x, y) ∂ν)) =ᶠ[ae μ] fun x => F (∫ (y : β), f (x, y) ∂ν + ∫ (y : β), g (x, y) ∂ν) ** filter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → E' hf : Integrable f hg : Integrable g a✝ : α h2f : Integrable fun y => f (a✝, y) h2g : Integrable fun y => g (a✝, y) ⊢ F (∫ (y : β), f (a✝, y) + g (a✝, y) ∂ν) = F (∫ (y : β), f (a✝, y) ∂ν + ∫ (y : β), g (a✝, y) ∂ν) ** simp [integral_add h2f h2g] ** Qed
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MeasureTheory.integral_fn_integral_sub ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → E' hf : Integrable f hg : Integrable g ⊢ ∫ (x : α), F (∫ (y : β), f (x, y) - g (x, y) ∂ν) ∂μ = ∫ (x : α), F (∫ (y : β), f (x, y) ∂ν - ∫ (y : β), g (x, y) ∂ν) ∂μ ** refine' integral_congr_ae _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → E' hf : Integrable f hg : Integrable g ⊢ (fun x => F (∫ (y : β), f (x, y) - g (x, y) ∂ν)) =ᶠ[ae μ] fun x => F (∫ (y : β), f (x, y) ∂ν - ∫ (y : β), g (x, y) ∂ν) ** filter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → E' hf : Integrable f hg : Integrable g a✝ : α h2f : Integrable fun y => f (a✝, y) h2g : Integrable fun y => g (a✝, y) ⊢ F (∫ (y : β), f (a✝, y) - g (a✝, y) ∂ν) = F (∫ (y : β), f (a✝, y) ∂ν - ∫ (y : β), g (a✝, y) ∂ν) ** simp [integral_sub h2f h2g] ** Qed
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MeasureTheory.lintegral_fn_integral_sub ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → ℝ≥0∞ hf : Integrable f hg : Integrable g ⊢ ∫⁻ (x : α), F (∫ (y : β), f (x, y) - g (x, y) ∂ν) ∂μ = ∫⁻ (x : α), F (∫ (y : β), f (x, y) ∂ν - ∫ (y : β), g (x, y) ∂ν) ∂μ ** refine' lintegral_congr_ae _ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → ℝ≥0∞ hf : Integrable f hg : Integrable g ⊢ (fun x => F (∫ (y : β), f (x, y) - g (x, y) ∂ν)) =ᶠ[ae μ] fun x => F (∫ (y : β), f (x, y) ∂ν - ∫ (y : β), g (x, y) ∂ν) ** filter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g ** case h α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f g : α × β → E F : E → ℝ≥0∞ hf : Integrable f hg : Integrable g a✝ : α h2f : Integrable fun y => f (a✝, y) h2g : Integrable fun y => g (a✝, y) ⊢ F (∫ (y : β), f (a✝, y) - g (a✝, y) ∂ν) = F (∫ (y : β), f (a✝, y) ∂ν - ∫ (y : β), g (a✝, y) ∂ν) ** simp [integral_sub h2f h2g] ** Qed
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MeasureTheory.integral_prod_symm ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f : α × β → E hf : Integrable f ⊢ ∫ (z : α × β), f z ∂Measure.prod μ ν = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν ** rw [← integral_prod_swap f] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f : α × β → E hf : Integrable f ⊢ ∫ (z : β × α), f (Prod.swap z) ∂Measure.prod ν μ = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν ** exact integral_prod _ hf.swap ** Qed
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MeasureTheory.set_integral_prod ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f : α × β → E s : Set α t : Set β hf : IntegrableOn f (s ×ˢ t) ⊢ ∫ (z : α × β) in s ×ˢ t, f z ∂Measure.prod μ ν = ∫ (x : α) in s, ∫ (y : β) in t, f (x, y) ∂ν ∂μ ** simp only [← Measure.prod_restrict s t, IntegrableOn] at hf ⊢ ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f : α × β → E s : Set α t : Set β hf : Integrable f ⊢ ∫ (z : α × β), f z ∂Measure.prod (Measure.restrict μ s) (Measure.restrict ν t) = ∫ (x : α) in s, ∫ (y : β) in t, f (x, y) ∂ν ∂μ ** exact integral_prod f hf ** Qed
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MeasureTheory.set_integral_prod_mul ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : MeasurableSpace α' inst✝⁹ : MeasurableSpace β inst✝⁸ : MeasurableSpace β' inst✝⁷ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁶ : NormedAddCommGroup E inst✝⁵ : SigmaFinite ν inst✝⁴ : NormedSpace ℝ E inst✝³ : SigmaFinite μ E' : Type u_7 inst✝² : NormedAddCommGroup E' inst✝¹ : NormedSpace ℝ E' L : Type u_8 inst✝ : IsROrC L f : α → L g : β → L s : Set α t : Set β ⊢ ∫ (z : α × β) in s ×ˢ t, f z.1 * g z.2 ∂Measure.prod μ ν = (∫ (x : α) in s, f x ∂μ) * ∫ (y : β) in t, g y ∂ν ** rw [← Measure.prod_restrict s t] ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : MeasurableSpace α' inst✝⁹ : MeasurableSpace β inst✝⁸ : MeasurableSpace β' inst✝⁷ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁶ : NormedAddCommGroup E inst✝⁵ : SigmaFinite ν inst✝⁴ : NormedSpace ℝ E inst✝³ : SigmaFinite μ E' : Type u_7 inst✝² : NormedAddCommGroup E' inst✝¹ : NormedSpace ℝ E' L : Type u_8 inst✝ : IsROrC L f : α → L g : β → L s : Set α t : Set β ⊢ ∫ (z : α × β), f z.1 * g z.2 ∂Measure.prod (Measure.restrict μ s) (Measure.restrict ν t) = (∫ (x : α) in s, f x ∂μ) * ∫ (y : β) in t, g y ∂ν ** apply integral_prod_mul ** Qed
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MeasureTheory.integral_fun_snd ** α : Type u_1 α' : Type u_2 β : Type u_3 β' : Type u_4 γ : Type u_5 E : Type u_6 inst✝¹⁰ : MeasurableSpace α inst✝⁹ : MeasurableSpace α' inst✝⁸ : MeasurableSpace β inst✝⁷ : MeasurableSpace β' inst✝⁶ : MeasurableSpace γ μ μ' : Measure α ν ν' : Measure β τ : Measure γ inst✝⁵ : NormedAddCommGroup E inst✝⁴ : SigmaFinite ν inst✝³ : NormedSpace ℝ E inst✝² : SigmaFinite μ E' : Type u_7 inst✝¹ : NormedAddCommGroup E' inst✝ : NormedSpace ℝ E' f : β → E ⊢ ∫ (z : α × β), f z.2 ∂Measure.prod μ ν = ENNReal.toReal (↑↑μ univ) • ∫ (y : β), f y ∂ν ** simpa using integral_prod_smul (1 : α → ℝ) f ** Qed
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MeasureTheory.LocallyIntegrableOn.exists_countable_integrableOn ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s ⊢ ∃ T, Set.Countable T ∧ (∀ (u : Set X), u ∈ T → IsOpen u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) ** have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ IntegrableOn f (u ∩ s) μ := by
rintro ⟨x, hx⟩
rcases hf x hx with ⟨t, ht, h't⟩
rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
refine' ⟨u, u_open, x_mem, h't.mono_set u_sub⟩ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s this : ∀ (x : ↑s), ∃ u, IsOpen u ∧ ↑x ∈ u ∧ IntegrableOn f (u ∩ s) ⊢ ∃ T, Set.Countable T ∧ (∀ (u : Set X), u ∈ T → IsOpen u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) ** choose u u_open xu hu using this ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) ⊢ ∃ T, Set.Countable T ∧ (∀ (u : Set X), u ∈ T → IsOpen u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) ** obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T.Countable ∧ s ⊆ ⋃ i ∈ T, u i := by
have : s ⊆ ⋃ x : s, u x := fun y hy => mem_iUnion_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
obtain ⟨T, hT_count, hT_un⟩ := isOpen_iUnion_countable u u_open
exact ⟨T, hT_count, by rwa [hT_un]⟩ ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) T : Set ↑s T_count : Set.Countable T hT : s ⊆ ⋃ i ∈ T, u i ⊢ ∃ T, Set.Countable T ∧ (∀ (u : Set X), u ∈ T → IsOpen u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) ** refine' ⟨u '' T, T_count.image _, _, by rwa [biUnion_image], _⟩ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s ⊢ ∀ (x : ↑s), ∃ u, IsOpen u ∧ ↑x ∈ u ∧ IntegrableOn f (u ∩ s) ** rintro ⟨x, hx⟩ ** case mk X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s x : X hx : x ∈ s ⊢ ∃ u, IsOpen u ∧ ↑{ val := x, property := hx } ∈ u ∧ IntegrableOn f (u ∩ s) ** rcases hf x hx with ⟨t, ht, h't⟩ ** case mk.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s x : X hx : x ∈ s t : Set X ht : t ∈ 𝓝[s] x h't : IntegrableOn f t ⊢ ∃ u, IsOpen u ∧ ↑{ val := x, property := hx } ∈ u ∧ IntegrableOn f (u ∩ s) ** rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩ ** case mk.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s x : X hx : x ∈ s t : Set X ht : t ∈ 𝓝[s] x h't : IntegrableOn f t u : Set X u_open : IsOpen u x_mem : x ∈ u u_sub : u ∩ s ⊆ t ⊢ ∃ u, IsOpen u ∧ ↑{ val := x, property := hx } ∈ u ∧ IntegrableOn f (u ∩ s) ** refine' ⟨u, u_open, x_mem, h't.mono_set u_sub⟩ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) ⊢ ∃ T, Set.Countable T ∧ s ⊆ ⋃ i ∈ T, u i ** have : s ⊆ ⋃ x : s, u x := fun y hy => mem_iUnion_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩) ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) this : s ⊆ ⋃ x, u x ⊢ ∃ T, Set.Countable T ∧ s ⊆ ⋃ i ∈ T, u i ** obtain ⟨T, hT_count, hT_un⟩ := isOpen_iUnion_countable u u_open ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) this : s ⊆ ⋃ x, u x T : Set ↑s hT_count : Set.Countable T hT_un : ⋃ i ∈ T, u i = ⋃ i, u i ⊢ ∃ T, Set.Countable T ∧ s ⊆ ⋃ i ∈ T, u i ** exact ⟨T, hT_count, by rwa [hT_un]⟩ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) this : s ⊆ ⋃ x, u x T : Set ↑s hT_count : Set.Countable T hT_un : ⋃ i ∈ T, u i = ⋃ i, u i ⊢ s ⊆ ⋃ i ∈ T, u i ** rwa [hT_un] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) T : Set ↑s T_count : Set.Countable T hT : s ⊆ ⋃ i ∈ T, u i ⊢ s ⊆ ⋃ u_1 ∈ u '' T, u_1 ** rwa [biUnion_image] ** case intro.intro.refine'_1 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) T : Set ↑s T_count : Set.Countable T hT : s ⊆ ⋃ i ∈ T, u i ⊢ ∀ (u_1 : Set X), u_1 ∈ u '' T → IsOpen u_1 ** rintro v ⟨w, -, rfl⟩ ** case intro.intro.refine'_1.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) T : Set ↑s T_count : Set.Countable T hT : s ⊆ ⋃ i ∈ T, u i w : ↑s ⊢ IsOpen (u w) ** exact u_open _ ** case intro.intro.refine'_2 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) T : Set ↑s T_count : Set.Countable T hT : s ⊆ ⋃ i ∈ T, u i ⊢ ∀ (u_1 : Set X), u_1 ∈ u '' T → IntegrableOn f (u_1 ∩ s) ** rintro v ⟨w, -, rfl⟩ ** case intro.intro.refine'_2.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ↑s → Set X u_open : ∀ (x : ↑s), IsOpen (u x) xu : ∀ (x : ↑s), ↑x ∈ u x hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) T : Set ↑s T_count : Set.Countable T hT : s ⊆ ⋃ i ∈ T, u i w : ↑s ⊢ IntegrableOn f (u w ∩ s) ** exact hu _ ** Qed
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MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s ⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ** rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩ ** case intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) ⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ** let T' : Set (Set X) := insert ∅ T ** case intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T ⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ** have T'_count : T'.Countable := Countable.insert ∅ T_count ** case intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' ⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ** have T'_ne : T'.Nonempty := by simp only [insert_nonempty] ** case intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' ⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ** rcases T'_count.exists_eq_range T'_ne with ⟨u, hu⟩ ** case intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u ⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ** refine' ⟨u, _, _, _⟩ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' ⊢ Set.Nonempty T' ** simp only [insert_nonempty] ** case intro.intro.intro.intro.intro.refine'_1 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u ⊢ ∀ (n : ℕ), IsOpen (u n) ** intro n ** case intro.intro.intro.intro.intro.refine'_1 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ ⊢ IsOpen (u n) ** have : u n ∈ T' := by rw [hu]; exact mem_range_self n ** case intro.intro.intro.intro.intro.refine'_1 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ this : u n ∈ T' ⊢ IsOpen (u n) ** rcases mem_insert_iff.1 this with h|h ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ ⊢ u n ∈ T' ** rw [hu] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ ⊢ u n ∈ range u ** exact mem_range_self n ** case intro.intro.intro.intro.intro.refine'_1.inl X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ this : u n ∈ T' h : u n = ∅ ⊢ IsOpen (u n) ** rw [h] ** case intro.intro.intro.intro.intro.refine'_1.inl X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ this : u n ∈ T' h : u n = ∅ ⊢ IsOpen ∅ ** exact isOpen_empty ** case intro.intro.intro.intro.intro.refine'_1.inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ this : u n ∈ T' h : u n ∈ T ⊢ IsOpen (u n) ** exact T_open _ h ** case intro.intro.intro.intro.intro.refine'_2 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u ⊢ s ⊆ ⋃ n, u n ** intro x hx ** case intro.intro.intro.intro.intro.refine'_2 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s ⊢ x ∈ ⋃ n, u n ** obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx ** case intro.intro.intro.intro.intro.refine'_2.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s v : Set X hv : v ∈ T h'v : x ∈ v ⊢ x ∈ ⋃ n, u n ** have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv ** case intro.intro.intro.intro.intro.refine'_2.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s v : Set X hv : v ∈ T h'v : x ∈ v this : v ∈ range u ⊢ x ∈ ⋃ n, u n ** obtain ⟨n, rfl⟩ : ∃ n, u n = v := by simpa only [mem_range] using this ** case intro.intro.intro.intro.intro.refine'_2.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s n : ℕ hv : u n ∈ T h'v : x ∈ u n this : u n ∈ range u ⊢ x ∈ ⋃ n, u n ** exact mem_iUnion_of_mem _ h'v ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s ⊢ ∃ v, v ∈ T ∧ x ∈ v ** simpa only [mem_iUnion, exists_prop] using sT hx ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s v : Set X hv : v ∈ T h'v : x ∈ v ⊢ v ∈ range u ** rw [← hu] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s v : Set X hv : v ∈ T h'v : x ∈ v ⊢ v ∈ T' ** exact subset_insert ∅ T hv ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u x : X hx : x ∈ s v : Set X hv : v ∈ T h'v : x ∈ v this : v ∈ range u ⊢ ∃ n, u n = v ** simpa only [mem_range] using this ** case intro.intro.intro.intro.intro.refine'_3 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u ⊢ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ** intro n ** case intro.intro.intro.intro.intro.refine'_3 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ ⊢ IntegrableOn f (u n ∩ s) ** have : u n ∈ T' := by rw [hu]; exact mem_range_self n ** case intro.intro.intro.intro.intro.refine'_3 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ this : u n ∈ T' ⊢ IntegrableOn f (u n ∩ s) ** rcases mem_insert_iff.1 this with h|h ** case intro.intro.intro.intro.intro.refine'_3.inl X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ this : u n ∈ T' h : u n = ∅ ⊢ IntegrableOn f (u n ∩ s) ** simp only [h, empty_inter, integrableOn_empty] ** case intro.intro.intro.intro.intro.refine'_3.inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s T : Set (Set X) T_count : Set.Countable T T_open : ∀ (u : Set X), u ∈ T → IsOpen u sT : s ⊆ ⋃ u ∈ T, u hT : ∀ (u : Set X), u ∈ T → IntegrableOn f (u ∩ s) T' : Set (Set X) := insert ∅ T T'_count : Set.Countable T' T'_ne : Set.Nonempty T' u : ℕ → Set X hu : T' = range u n : ℕ this : u n ∈ T' h : u n ∈ T ⊢ IntegrableOn f (u n ∩ s) ** exact hT _ h ** Qed
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MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s ⊢ AEStronglyMeasurable f (Measure.restrict μ s) ** rcases hf.exists_nat_integrableOn with ⟨u, -, su, hu⟩ ** case intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ℕ → Set X su : s ⊆ ⋃ n, u n hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ⊢ AEStronglyMeasurable f (Measure.restrict μ s) ** have : s = ⋃ n, u n ∩ s := by rw [← iUnion_inter]; exact (inter_eq_right.mpr su).symm ** case intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ℕ → Set X su : s ⊆ ⋃ n, u n hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) this : s = ⋃ n, u n ∩ s ⊢ AEStronglyMeasurable f (Measure.restrict μ s) ** rw [this, aestronglyMeasurable_iUnion_iff] ** case intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ℕ → Set X su : s ⊆ ⋃ n, u n hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) this : s = ⋃ n, u n ∩ s ⊢ ∀ (i : ℕ), AEStronglyMeasurable f (Measure.restrict μ (u i ∩ s)) ** exact fun i : ℕ => (hu i).aestronglyMeasurable ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ℕ → Set X su : s ⊆ ⋃ n, u n hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ⊢ s = ⋃ n, u n ∩ s ** rw [← iUnion_inter] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrableOn f s u : ℕ → Set X su : s ⊆ ⋃ n, u n hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) ⊢ s = (⋃ i, u i) ∩ s ** exact (inter_eq_right.mpr su).symm ** Qed
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MeasureTheory.locallyIntegrableOn_iff ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁶ : MeasurableSpace X inst✝⁵ : TopologicalSpace X inst✝⁴ : MeasurableSpace Y inst✝³ : TopologicalSpace Y inst✝² : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝¹ : LocallyCompactSpace X inst✝ : T2Space X hs : IsClosed s ∨ IsOpen s ⊢ LocallyIntegrableOn f s ↔ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k ** refine' ⟨fun hf k hk => hf.integrableOn_compact_subset hk, fun hf x hx => _⟩ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁶ : MeasurableSpace X inst✝⁵ : TopologicalSpace X inst✝⁴ : MeasurableSpace Y inst✝³ : TopologicalSpace Y inst✝² : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝¹ : LocallyCompactSpace X inst✝ : T2Space X hs : IsClosed s ∨ IsOpen s hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k x : X hx : x ∈ s ⊢ IntegrableAtFilter f (𝓝[s] x) ** cases hs with
| inl hs =>
exact
let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
⟨_, inter_mem_nhdsWithin s h2K,
hf _ (inter_subset_left _ _)
(hK.of_isClosed_subset (hs.inter hK.isClosed) (inter_subset_right _ _))⟩
| inr hs =>
obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
refine' ⟨K, _, hf K h3K hK⟩
simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K ** case inl X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁶ : MeasurableSpace X inst✝⁵ : TopologicalSpace X inst✝⁴ : MeasurableSpace Y inst✝³ : TopologicalSpace Y inst✝² : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝¹ : LocallyCompactSpace X inst✝ : T2Space X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k x : X hx : x ∈ s hs : IsClosed s ⊢ IntegrableAtFilter f (𝓝[s] x) ** exact
let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
⟨_, inter_mem_nhdsWithin s h2K,
hf _ (inter_subset_left _ _)
(hK.of_isClosed_subset (hs.inter hK.isClosed) (inter_subset_right _ _))⟩ ** case inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁶ : MeasurableSpace X inst✝⁵ : TopologicalSpace X inst✝⁴ : MeasurableSpace Y inst✝³ : TopologicalSpace Y inst✝² : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝¹ : LocallyCompactSpace X inst✝ : T2Space X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k x : X hx : x ∈ s hs : IsOpen s ⊢ IntegrableAtFilter f (𝓝[s] x) ** obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx ** case inr.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁶ : MeasurableSpace X inst✝⁵ : TopologicalSpace X inst✝⁴ : MeasurableSpace Y inst✝³ : TopologicalSpace Y inst✝² : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝¹ : LocallyCompactSpace X inst✝ : T2Space X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k x : X hx : x ∈ s hs : IsOpen s K : Set X hK : IsCompact K h2K : x ∈ interior K h3K : K ⊆ s ⊢ IntegrableAtFilter f (𝓝[s] x) ** refine' ⟨K, _, hf K h3K hK⟩ ** case inr.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁶ : MeasurableSpace X inst✝⁵ : TopologicalSpace X inst✝⁴ : MeasurableSpace Y inst✝³ : TopologicalSpace Y inst✝² : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝¹ : LocallyCompactSpace X inst✝ : T2Space X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k x : X hx : x ∈ s hs : IsOpen s K : Set X hK : IsCompact K h2K : x ∈ interior K h3K : K ⊆ s ⊢ K ∈ 𝓝[s] x ** simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K ** Qed
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MeasureTheory.locallyIntegrableOn_univ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X ⊢ LocallyIntegrableOn f univ ↔ LocallyIntegrable f ** simp only [LocallyIntegrableOn, nhdsWithin_univ, mem_univ, true_imp_iff] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X ⊢ (∀ (x : X), IntegrableAtFilter f (𝓝 x)) ↔ LocallyIntegrable f ** rfl ** Qed
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MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : OpensMeasurableSpace X hf : LocallyIntegrable f ⊢ LocallyIntegrableOn f s ** intro x _ ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : OpensMeasurableSpace X hf : LocallyIntegrable f x : X a✝ : x ∈ s ⊢ IntegrableAtFilter f (𝓝[s] x) ** obtain ⟨t, ht_mem, ht_int⟩ := hf x ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : OpensMeasurableSpace X hf : LocallyIntegrable f x : X a✝ : x ∈ s t : Set X ht_mem : t ∈ 𝓝 x ht_int : IntegrableOn f t ⊢ IntegrableAtFilter f (𝓝[s] x) ** obtain ⟨u, hu_sub, hu_o, hu_mem⟩ := mem_nhds_iff.mp ht_mem ** case intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : OpensMeasurableSpace X hf : LocallyIntegrable f x : X a✝ : x ∈ s t : Set X ht_mem : t ∈ 𝓝 x ht_int : IntegrableOn f t u : Set X hu_sub : u ⊆ t hu_o : IsOpen u hu_mem : x ∈ u ⊢ IntegrableAtFilter f (𝓝[s] x) ** refine' ⟨_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), _⟩ ** case intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : OpensMeasurableSpace X hf : LocallyIntegrable f x : X a✝ : x ∈ s t : Set X ht_mem : t ∈ 𝓝 x ht_int : IntegrableOn f t u : Set X hu_sub : u ⊆ t hu_o : IsOpen u hu_mem : x ∈ u ⊢ IntegrableOn f (s ∩ u) ** simpa only [IntegrableOn, Measure.restrict_restrict hu_o.measurableSet, inter_comm] using
ht_int.mono_set hu_sub ** Qed
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MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k ⊢ ∃ u, IsOpen u ∧ k ⊆ u ∧ IntegrableOn f u ** refine' IsCompact.induction_on hk _ _ _ _ ** case refine'_1 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k ⊢ ∃ u, IsOpen u ∧ ∅ ⊆ u ∧ IntegrableOn f u ** refine' ⟨∅, isOpen_empty, Subset.rfl, integrableOn_empty⟩ ** case refine'_2 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k ⊢ ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u) → ∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u ** rintro s t hst ⟨u, u_open, tu, hu⟩ ** case refine'_2.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s✝ : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k s t : Set X hst : s ⊆ t u : Set X u_open : IsOpen u tu : t ⊆ u hu : IntegrableOn f u ⊢ ∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u ** exact ⟨u, u_open, hst.trans tu, hu⟩ ** case refine'_3 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k ⊢ ∀ ⦃s t : Set X⦄, (∃ u, IsOpen u ∧ s ⊆ u ∧ IntegrableOn f u) → (∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u) → ∃ u, IsOpen u ∧ s ∪ t ⊆ u ∧ IntegrableOn f u ** rintro s t ⟨u, u_open, su, hu⟩ ⟨v, v_open, tv, hv⟩ ** case refine'_3.intro.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s✝ : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k s t u : Set X u_open : IsOpen u su : s ⊆ u hu : IntegrableOn f u v : Set X v_open : IsOpen v tv : t ⊆ v hv : IntegrableOn f v ⊢ ∃ u, IsOpen u ∧ s ∪ t ⊆ u ∧ IntegrableOn f u ** exact ⟨u ∪ v, u_open.union v_open, union_subset_union su tv, hu.union hv⟩ ** case refine'_4 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k ⊢ ∀ (x : X), x ∈ k → ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u ** intro x _ ** case refine'_4 X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k x : X a✝ : x ∈ k ⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u ** rcases hf x with ⟨u, ux, hu⟩ ** case refine'_4.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k x : X a✝ : x ∈ k u : Set X ux : u ∈ 𝓝 x hu : IntegrableOn f u ⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u ** rcases mem_nhds_iff.1 ux with ⟨v, vu, v_open, xv⟩ ** case refine'_4.intro.intro.intro.intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X hf : LocallyIntegrable f k : Set X hk : IsCompact k x : X a✝ : x ∈ k u : Set X ux : u ∈ 𝓝 x hu : IntegrableOn f u v : Set X vu : v ⊆ u v_open : IsOpen v xv : x ∈ v ⊢ ∃ t, t ∈ 𝓝[k] x ∧ ∃ u, IsOpen u ∧ t ⊆ u ∧ IntegrableOn f u ** exact ⟨v, nhdsWithin_le_nhds (v_open.mem_nhds xv), v, v_open, Subset.rfl, hu.mono_set vu⟩ ** Qed
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MeasureTheory.LocallyIntegrable.aestronglyMeasurable ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝ : SecondCountableTopology X hf : LocallyIntegrable f ⊢ AEStronglyMeasurable f μ ** simpa only [restrict_univ] using (locallyIntegrableOn_univ.mpr hf).aestronglyMeasurable ** Qed
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MeasureTheory.Memℒp.locallyIntegrable ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f✝ g : X → E μ : Measure X s : Set X inst✝ : IsLocallyFiniteMeasure μ f : X → E p : ℝ≥0∞ hf : Memℒp f p hp : 1 ≤ p ⊢ LocallyIntegrable f ** intro x ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f✝ g : X → E μ : Measure X s : Set X inst✝ : IsLocallyFiniteMeasure μ f : X → E p : ℝ≥0∞ hf : Memℒp f p hp : 1 ≤ p x : X ⊢ IntegrableAtFilter f (𝓝 x) ** rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩ ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f✝ g : X → E μ : Measure X s : Set X inst✝ : IsLocallyFiniteMeasure μ f : X → E p : ℝ≥0∞ hf : Memℒp f p hp : 1 ≤ p x : X U : Set X hU : U ∈ 𝓝 x h'U : ↑↑μ U < ⊤ ⊢ IntegrableAtFilter f (𝓝 x) ** have : Fact (μ U < ⊤) := ⟨h'U⟩ ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f✝ g : X → E μ : Measure X s : Set X inst✝ : IsLocallyFiniteMeasure μ f : X → E p : ℝ≥0∞ hf : Memℒp f p hp : 1 ≤ p x : X U : Set X hU : U ∈ 𝓝 x h'U : ↑↑μ U < ⊤ this : Fact (↑↑μ U < ⊤) ⊢ IntegrableAtFilter f (𝓝 x) ** refine' ⟨U, hU, _⟩ ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f✝ g : X → E μ : Measure X s : Set X inst✝ : IsLocallyFiniteMeasure μ f : X → E p : ℝ≥0∞ hf : Memℒp f p hp : 1 ≤ p x : X U : Set X hU : U ∈ 𝓝 x h'U : ↑↑μ U < ⊤ this : Fact (↑↑μ U < ⊤) ⊢ IntegrableOn f U ** rw [IntegrableOn, ← memℒp_one_iff_integrable] ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : TopologicalSpace X inst✝³ : MeasurableSpace Y inst✝² : TopologicalSpace Y inst✝¹ : NormedAddCommGroup E f✝ g : X → E μ : Measure X s : Set X inst✝ : IsLocallyFiniteMeasure μ f : X → E p : ℝ≥0∞ hf : Memℒp f p hp : 1 ≤ p x : X U : Set X hU : U ∈ 𝓝 x h'U : ↑↑μ U < ⊤ this : Fact (↑↑μ U < ⊤) ⊢ Memℒp f 1 ** apply (hf.restrict U).memℒp_of_exponent_le hp ** Qed
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MeasureTheory.LocallyIntegrable.indicator ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s✝ : Set X hf : LocallyIntegrable f s : Set X hs : MeasurableSet s ⊢ LocallyIntegrable (Set.indicator s f) ** intro x ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s✝ : Set X hf : LocallyIntegrable f s : Set X hs : MeasurableSet s x : X ⊢ IntegrableAtFilter (Set.indicator s f) (𝓝 x) ** rcases hf x with ⟨U, hU, h'U⟩ ** case intro.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁴ : MeasurableSpace X inst✝³ : TopologicalSpace X inst✝² : MeasurableSpace Y inst✝¹ : TopologicalSpace Y inst✝ : NormedAddCommGroup E f g : X → E μ : Measure X s✝ : Set X hf : LocallyIntegrable f s : Set X hs : MeasurableSet s x : X U : Set X hU : U ∈ 𝓝 x h'U : IntegrableOn f U ⊢ IntegrableAtFilter (Set.indicator s f) (𝓝 x) ** exact ⟨U, hU, h'U.indicator hs⟩ ** Qed
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MeasureTheory.LocallyIntegrable.integrable_smul_left_of_hasCompactSupport ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g ⊢ Integrable fun x => g x • f x ** let K := tsupport g ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g ⊢ Integrable fun x => g x • f x ** have hK : IsCompact K := h'g ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K ⊢ Integrable fun x => g x • f x ** have : K.indicator (fun x ↦ g x • f x) = (fun x ↦ g x • f x) := by
apply indicator_eq_self.2
apply support_subset_iff'.2
intros x hx
simp [image_eq_zero_of_nmem_tsupport hx] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x ⊢ Integrable fun x => g x • f x ** rw [← this, indicator_smul] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x ⊢ Integrable fun x => g x • Set.indicator K (fun x => f x) x ** apply Integrable.smul_of_top_right ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K ⊢ (Set.indicator K fun x => g x • f x) = fun x => g x • f x ** apply indicator_eq_self.2 ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K ⊢ (support fun x => g x • f x) ⊆ K ** apply support_subset_iff'.2 ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K ⊢ ∀ (x : X), ¬x ∈ K → g x • f x = 0 ** intros x hx ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K x : X hx : ¬x ∈ K ⊢ g x • f x = 0 ** simp [image_eq_zero_of_nmem_tsupport hx] ** case hf X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x ⊢ Integrable fun x => Set.indicator K (fun x => f x) x ** rw [integrable_indicator_iff hK.measurableSet] ** case hf X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x ⊢ IntegrableOn (fun x => f x) K ** exact hf.integrableOn_isCompact hK ** case hφ X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g✝ : X → E μ : Measure X s : Set X inst✝² : NormedSpace ℝ E inst✝¹ : OpensMeasurableSpace X inst✝ : T2Space X hf : LocallyIntegrable f g : X → ℝ hg : Continuous g h'g : HasCompactSupport g K : Set X := tsupport g hK : IsCompact K this : (Set.indicator K fun x => g x • f x) = fun x => g x • f x ⊢ Memℒp (fun x => g x) ⊤ ** exact hg.memℒp_top_of_hasCompactSupport h'g μ ** Qed
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ContinuousOn.integrableOn_compact ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝² : OpensMeasurableSpace X inst✝¹ : IsLocallyFiniteMeasure μ K : Set X a b : X inst✝ : MetrizableSpace X hK : IsCompact K hf : ContinuousOn f K ⊢ IntegrableOn f K ** letI := metrizableSpaceMetric X ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝² : OpensMeasurableSpace X inst✝¹ : IsLocallyFiniteMeasure μ K : Set X a b : X inst✝ : MetrizableSpace X hK : IsCompact K hf : ContinuousOn f K this : MetricSpace X := metrizableSpaceMetric X ⊢ IntegrableOn f K ** refine' LocallyIntegrableOn.integrableOn_isCompact (fun x hx => _) hK ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁷ : MeasurableSpace X inst✝⁶ : TopologicalSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : TopologicalSpace Y inst✝³ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝² : OpensMeasurableSpace X inst✝¹ : IsLocallyFiniteMeasure μ K : Set X a b : X inst✝ : MetrizableSpace X hK : IsCompact K hf : ContinuousOn f K this : MetricSpace X := metrizableSpaceMetric X x : X hx : x ∈ K ⊢ IntegrableAtFilter f (𝓝[K] x) ** exact hf.integrableAt_nhdsWithin_of_isSeparable hK.measurableSet hK.isSeparable hx ** Qed
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MonotoneOn.integrableOn_of_measure_ne_top ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s ⊢ IntegrableOn f s ** borelize E ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E ⊢ IntegrableOn f s ** obtain rfl | _ := s.eq_empty_or_nonempty ** case inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E h✝ : Set.Nonempty s ⊢ IntegrableOn f s ** have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩ ** case inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E h✝ : Set.Nonempty s hbelow : BddBelow (f '' s) ⊢ IntegrableOn f s ** have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩ ** case inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E h✝ : Set.Nonempty s hbelow : BddBelow (f '' s) habove : BddAbove (f '' s) ⊢ IntegrableOn f s ** have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow ** case inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E h✝ : Set.Nonempty s hbelow : BddBelow (f '' s) habove : BddAbove (f '' s) this : IsBounded (f '' s) ⊢ IntegrableOn f s ** rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩ ** case inr.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E h✝ : Set.Nonempty s hbelow : BddBelow (f '' s) habove : BddAbove (f '' s) this : IsBounded (f '' s) C : ℝ hC : ∀ (x : E), x ∈ f '' s → ‖x‖ ≤ C ⊢ IntegrableOn f s ** have A : IntegrableOn (fun _ => C) s μ := by
simp only [hs.lt_top, integrableOn_const, or_true_iff] ** case inr.intro X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E h✝ : Set.Nonempty s hbelow : BddBelow (f '' s) habove : BddAbove (f '' s) this : IsBounded (f '' s) C : ℝ hC : ∀ (x : E), x ∈ f '' s → ‖x‖ ≤ C A : IntegrableOn (fun x => C) s ⊢ IntegrableOn f s ** refine'
Integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).aestronglyMeasurable
((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy)) ** case inl X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E a b : X this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E hmono : MonotoneOn f ∅ ha : IsLeast ∅ a hb : IsGreatest ∅ b hs : ↑↑μ ∅ ≠ ⊤ h's : MeasurableSet ∅ ⊢ IntegrableOn f ∅ ** exact integrableOn_empty ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹⁰ : MeasurableSpace X inst✝⁹ : TopologicalSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : TopologicalSpace Y inst✝⁶ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁵ : BorelSpace X inst✝⁴ : ConditionallyCompleteLinearOrder X inst✝³ : ConditionallyCompleteLinearOrder E inst✝² : OrderTopology X inst✝¹ : OrderTopology E inst✝ : SecondCountableTopology E hmono : MonotoneOn f s a b : X ha : IsLeast s a hb : IsGreatest s b hs : ↑↑μ s ≠ ⊤ h's : MeasurableSet s this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E h✝ : Set.Nonempty s hbelow : BddBelow (f '' s) habove : BddAbove (f '' s) this : IsBounded (f '' s) C : ℝ hC : ∀ (x : E), x ∈ f '' s → ‖x‖ ≤ C ⊢ IntegrableOn (fun x => C) s ** simp only [hs.lt_top, integrableOn_const, or_true_iff] ** Qed
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