Datasets:

akjadhav
/

leandojo-lean4-formal-informal-strings-split

Dataset card Data Studio Files Files and versions Community

Split (3)

train · 47.4k rows

formal stringlengths 41 427k	informal stringclasses 1 value
MonotoneOn.integrableOn_isCompact X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹¹ : MeasurableSpace X inst✝¹⁰ : TopologicalSpace X inst✝⁹ : MeasurableSpace Y inst✝⁸ : TopologicalSpace Y inst✝⁷ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁶ : BorelSpace X inst✝⁵ : ConditionallyCompleteLinearOrder X inst✝⁴ : ConditionallyCompleteLinearOrder E inst✝³ : OrderTopology X inst✝² : OrderTopology E inst✝¹ : SecondCountableTopology E inst✝ : IsFiniteMeasureOnCompacts μ hs : IsCompact s hmono : MonotoneOn f s ⊢ IntegrableOn f s obtain rfl \| h := s.eq_empty_or_nonempty case inl X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹¹ : MeasurableSpace X inst✝¹⁰ : TopologicalSpace X inst✝⁹ : MeasurableSpace Y inst✝⁸ : TopologicalSpace Y inst✝⁷ : NormedAddCommGroup E f g : X → E μ : Measure X inst✝⁶ : BorelSpace X inst✝⁵ : ConditionallyCompleteLinearOrder X inst✝⁴ : ConditionallyCompleteLinearOrder E inst✝³ : OrderTopology X inst✝² : OrderTopology E inst✝¹ : SecondCountableTopology E inst✝ : IsFiniteMeasureOnCompacts μ hs : IsCompact ∅ hmono : MonotoneOn f ∅ ⊢ IntegrableOn f ∅ exact integrableOn_empty case inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹¹ : MeasurableSpace X inst✝¹⁰ : TopologicalSpace X inst✝⁹ : MeasurableSpace Y inst✝⁸ : TopologicalSpace Y inst✝⁷ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁶ : BorelSpace X inst✝⁵ : ConditionallyCompleteLinearOrder X inst✝⁴ : ConditionallyCompleteLinearOrder E inst✝³ : OrderTopology X inst✝² : OrderTopology E inst✝¹ : SecondCountableTopology E inst✝ : IsFiniteMeasureOnCompacts μ hs : IsCompact s hmono : MonotoneOn f s h : Set.Nonempty s ⊢ IntegrableOn f s exact hmono.integrableOn_of_measure_ne_top (hs.isLeast_sInf h) (hs.isGreatest_sSup h) hs.measure_lt_top.ne hs.measurableSet ** Qed
MeasureTheory.IntegrableOn.mul_continuousOn_of_subset ** 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : IntegrableOn g A hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K ⊢ IntegrableOn (fun x => g x * g' x) A rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : IntegrableOn g A hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C ⊢ IntegrableOn (fun x => g x * g' x) A rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C ⊢ Memℒp (fun x => g x * g' x) 1 ** have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by filter_upwards [ae_restrict_mem hA] with x hx refine' (norm_mul_le _ _).trans _ rw [mul_comm] apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) ** case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖ ⊢ Memℒp (fun x => g x * g' x) 1 exact Memℒp.of_le_mul hg (hg.aestronglyMeasurable.mul <\| (hg'.mono hAK).aestronglyMeasurable hA) this 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C ⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖ filter_upwards [ae_restrict_mem hA] with x hx case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x * g' x‖ ≤ C * ‖g x‖ refine' (norm_mul_le _ _).trans _ case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g x‖ rw [mul_comm] case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g' x‖ * ‖g x‖ ≤ C * ‖g x‖ apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) Qed
MeasureTheory.IntegrableOn.continuousOn_mul_of_subset ** 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : IntegrableOn g' A hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K ⊢ IntegrableOn (fun x => g x * g' x) A rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : IntegrableOn g' A hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C ⊢ IntegrableOn (fun x => g x * g' x) A rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C ⊢ Memℒp (fun x => g x * g' x) 1 ** have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by filter_upwards [ae_restrict_mem hA] with x hx refine' (norm_mul_le _ _).trans _ apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) ** case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖ ⊢ Memℒp (fun x => g x * g' x) 1 exact Memℒp.of_le_mul hg' (((hg.mono hAK).aestronglyMeasurable hA).mul hg'.aestronglyMeasurable) this 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C ⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖ filter_upwards [ae_restrict_mem hA] with x hx case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖ refine' (norm_mul_le _ _).trans _ case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖ apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) Qed
MeasureTheory.IntegrableOn.continuousOn_smul X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ IntegrableOn (fun x => f x • g x) K rw [IntegrableOn, ← integrable_norm_iff] X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ Integrable fun a => ‖f a • g a‖ simp_rw [norm_smul] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ Integrable fun a => ‖f a‖ * ‖g a‖ refine' IntegrableOn.continuousOn_mul _ hg.norm hK X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ ContinuousOn (fun a => ‖f a‖) K exact continuous_norm.comp_continuousOn hf X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K) exact (hf.aestronglyMeasurable hK.measurableSet).smul hg.1 Qed
MeasureTheory.IntegrableOn.smul_continuousOn X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ IntegrableOn (fun x => f x • g x) K rw [IntegrableOn, ← integrable_norm_iff] X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ Integrable fun a => ‖f a • g a‖ simp_rw [norm_smul] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ Integrable fun a => ‖f a‖ * ‖g a‖ refine' IntegrableOn.mul_continuousOn hf.norm _ hK X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ ContinuousOn (fun a => ‖g a‖) K exact continuous_norm.comp_continuousOn hg X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K) exact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet) Qed
MeasureTheory.LocallyIntegrableOn.continuousOn_mul ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : LocallyIntegrableOn f s hg : ContinuousOn g s hs : IsOpen s ⊢ LocallyIntegrableOn (fun x => g x * f x) s rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢ X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k hg : ContinuousOn g s hs : IsOpen s ⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => g x * f x) k exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c Qed
MeasureTheory.LocallyIntegrableOn.mul_continuousOn ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : LocallyIntegrableOn f s hg : ContinuousOn g s hs : IsOpen s ⊢ LocallyIntegrableOn (fun x => f x * g x) s rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢ X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k hg : ContinuousOn g s hs : IsOpen s ⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => f x * g x) k exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c Qed
MeasureTheory.SimpleFunc.nnnorm_approxOn_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ have := edist_approxOn_le hf h₀ x n α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ rw [edist_comm y₀] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist (f x) y₀ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ simp only [edist_nndist, nndist_eq_nnnorm] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ↑‖f x - y₀‖₊ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ exact_mod_cast this ** Qed
MeasureTheory.SimpleFunc.norm_approxOn_y₀_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ have := edist_approxOn_y0_le hf h₀ x n α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist y₀ (↑(approxOn f hf s y₀ h₀ n) x) ≤ edist y₀ (f x) + edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ repeat' rw [edist_comm y₀, edist_eq_coe_nnnorm_sub] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖₊ ≤ ↑‖f x - y₀‖₊ + ↑‖f x - y₀‖₊ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ exact_mod_cast this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖₊ ≤ edist y₀ (f x) + edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ rw [edist_comm y₀, edist_eq_coe_nnnorm_sub] at this ** Qed
MeasureTheory.SimpleFunc.norm_approxOn_zero_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E h₀ : 0 ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ ⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖ have := edist_approxOn_y0_le hf h₀ x n α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E h₀ : 0 ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist 0 (↑(approxOn f hf s 0 h₀ n) x) ≤ edist 0 (f x) + edist 0 (f x) ⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖ simp [edist_comm (0 : E), edist_eq_coe_nnnorm] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E h₀ : 0 ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s 0 h₀ n) x‖₊ ≤ ↑‖f x‖₊ + ↑‖f x‖₊ ⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖ exact_mod_cast this ** Qed
MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_snorm α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ n : ℕ ⊢ 0 ∈ Set.range f ∪ {0} simp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ Tendsto (fun n => snorm (↑(approxOn f fmeas (Set.range f ∪ {0}) 0 (_ : 0 ∈ Set.range f ∪ {0}) n) - f) p μ) atTop (𝓝 0) refine' tendsto_approxOn_Lp_snorm fmeas _ hp_ne_top _ _ case refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ ∀ᵐ (x : β) ∂μ, f x ∈ closure (Set.range f ∪ {0}) apply eventually_of_forall case refine'_1.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ ∀ (x : β), f x ∈ closure (Set.range f ∪ {0}) intro x case refine'_1.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ x : β ⊢ f x ∈ closure (Set.range f ∪ {0}) apply subset_closure case refine'_1.hp.a α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ x : β ⊢ f x ∈ Set.range f ∪ {0} simp case refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ snorm (fun x => f x - 0) p μ < ⊤ simpa using hf ** Qed
MeasureTheory.SimpleFunc.memℒp_approxOn_range α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : Memℒp f p n : ℕ ⊢ 0 ∈ Set.range f ∪ {0} simp ** Qed
MeasureTheory.Memℒp.exists_simpleFunc_snorm_sub_lt α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p borelize E α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p let f' := hf.1.mk f α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p rsuffices ⟨g, hg, g_mem⟩ : ∃ g : β →ₛ E, snorm (f' - ⇑g) p μ < ε ∧ Memℒp g p μ α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p have hf' : Memℒp f' p μ := hf.ae_eq hf.1.ae_eq_mk α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p have f'meas : Measurable f' := hf.1.measurable_mk α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p have : SeparableSpace (range f' ∪ {0} : Set E) := StronglyMeasurable.separableSpace_range_union_singleton hf.1.stronglyMeasurable_mk α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' this : SeparableSpace ↑(Set.range f' ∪ {0}) ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p rcases ((tendsto_approxOn_range_Lp_snorm hp_ne_top f'meas hf'.2).eventually <\| gt_mem_nhds hε.bot_lt).exists with ⟨n, hn⟩ case intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' this : SeparableSpace ↑(Set.range f' ∪ {0}) n : ℕ hn : snorm (↑(approxOn f' f'meas (Set.range f' ∪ {0}) 0 (_ : 0 ∈ Set.range f' ∪ {0}) n) - f') p μ < ε ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p rw [← snorm_neg, neg_sub] at hn case intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' this : SeparableSpace ↑(Set.range f' ∪ {0}) n : ℕ hn : snorm (f' - ↑(approxOn f' f'meas (Set.range f' ∪ {0}) 0 (_ : 0 ∈ Set.range f' ∪ {0}) n)) p μ < ε ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p exact ⟨_, hn, memℒp_approxOn_range f'meas hf' _⟩ case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p refine' ⟨g, _, g_mem⟩ case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ snorm (f - ↑g) p μ < ε suffices snorm (f - ⇑g) p μ = snorm (f' - ⇑g) p μ by rwa [this] case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ snorm (f - ↑g) p μ = snorm (f' - ↑g) p μ apply snorm_congr_ae case intro.intro.hfg α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ f - ↑g =ᵐ[μ] f' - ↑g filter_upwards [hf.1.ae_eq_mk] with x hx case h α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p x : β hx : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x ⊢ (f - ↑g) x = (f' - ↑g) x simpa only [Pi.sub_apply, sub_left_inj] using hx α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p this : snorm (f - ↑g) p μ = snorm (f' - ↑g) p μ ⊢ snorm (f - ↑g) p μ < ε rwa [this] ** Qed
MeasureTheory.SimpleFunc.tendsto_approxOn_L1_nnnorm α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s μ : Measure β hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s hi : HasFiniteIntegral fun x => f x - y₀ ⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ∂μ) atTop (𝓝 0) simpa [snorm_one_eq_lintegral_nnnorm] using tendsto_approxOn_Lp_snorm hf h₀ one_ne_top hμ (by simpa [snorm_one_eq_lintegral_nnnorm] using hi) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s μ : Measure β hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s hi : HasFiniteIntegral fun x => f x - y₀ ⊢ snorm (fun x => f x - y₀) 1 μ < ⊤ simpa [snorm_one_eq_lintegral_nnnorm] using hi ** Qed
MeasureTheory.SimpleFunc.integrable_approxOn α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f hf : Integrable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s hi₀ : Integrable fun x => y₀ n : ℕ ⊢ Integrable ↑(approxOn f fmeas s y₀ h₀ n) rw [← memℒp_one_iff_integrable] at hf hi₀ ⊢ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f hf : Memℒp f 1 s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s hi₀ : Memℒp (fun x => y₀) 1 n : ℕ ⊢ Memℒp (↑(approxOn f fmeas s y₀ h₀ n)) 1 exact memℒp_approxOn fmeas hf h₀ hi₀ n ** Qed
MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_nnnorm α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f n : ℕ x : β ⊢ 0 ∈ Set.range f ∪ {0} simp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f fmeas (Set.range f ∪ {0}) 0 (_ : 0 ∈ Set.range f ∪ {0}) n) x - f x‖₊ ∂μ) atTop (𝓝 0) apply tendsto_approxOn_L1_nnnorm fmeas case hμ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ ∀ᵐ (x : β) ∂μ, f x ∈ closure (Set.range f ∪ {0}) apply eventually_of_forall case hμ.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ ∀ (x : β), f x ∈ closure (Set.range f ∪ {0}) intro x case hμ.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f x : β ⊢ f x ∈ closure (Set.range f ∪ {0}) apply subset_closure case hμ.hp.a α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f x : β ⊢ f x ∈ Set.range f ∪ {0} simp case hi α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ HasFiniteIntegral fun x => f x - 0 simpa using hf.2 ** Qed
MeasureTheory.SimpleFunc.integrable_approxOn_range α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : Integrable f n : ℕ ⊢ 0 ∈ Set.range f ∪ {0} simp ** Qed
MeasureTheory.SimpleFunc.snorm'_eq ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α ⊢ snorm' (↑f) p μ = (∑ y in SimpleFunc.range f, ↑‖y‖₊ ^ p * ↑↑μ (↑f ⁻¹' {y})) ^ (1 / p) have h_map : (fun a => (‖f a‖₊ : ℝ≥0∞) ^ p) = f.map fun a : F => (‖a‖₊ : ℝ≥0∞) ^ p := by simp; rfl α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α h_map : (fun a => ↑‖↑f a‖₊ ^ p) = ↑(map (fun a => ↑‖a‖₊ ^ p) f) ⊢ snorm' (↑f) p μ = (∑ y in SimpleFunc.range f, ↑‖y‖₊ ^ p * ↑↑μ (↑f ⁻¹' {y})) ^ (1 / p) rw [snorm', h_map, lintegral_eq_lintegral, map_lintegral] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α ⊢ (fun a => ↑‖↑f a‖₊ ^ p) = ↑(map (fun a => ↑‖a‖₊ ^ p) f) simp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α ⊢ (fun a => ↑‖↑f a‖₊ ^ p) = (fun a => ↑‖a‖₊ ^ p) ∘ ↑f rfl Qed
MeasureTheory.SimpleFunc.measure_support_lt_top α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ⊢ ↑↑μ (support ↑f) < ⊤ rw [support_eq] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ⊢ ↑↑μ (⋃ y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f), ↑f ⁻¹' {y}) < ⊤ refine' (measure_biUnion_finset_le _ _).trans_lt (ENNReal.sum_lt_top_iff.mpr fun y hy => _) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ y : β hy : y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f) ⊢ ↑↑μ (↑f ⁻¹' {y}) < ⊤ rw [Finset.mem_filter] at hy α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ y : β hy : y ∈ SimpleFunc.range f ∧ y ≠ 0 ⊢ ↑↑μ (↑f ⁻¹' {y}) < ⊤ exact hf y hy.2 ** Qed
MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : { x // x ∈ simpleFunc E p μ } ⊢ ↑(toSimpleFunc f) =ᵐ[μ] ↑↑↑f convert (AEEqFun.coeFn_mk (toSimpleFunc f) (toSimpleFunc f).aestronglyMeasurable).symm using 2 case h.e'_5.h.e'_6 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : { x // x ∈ simpleFunc E p μ } ⊢ ↑↑f = AEEqFun.mk ↑(toSimpleFunc f) (_ : AEStronglyMeasurable (↑(toSimpleFunc f)) μ) exact (Classical.choose_spec f.2).symm ** Qed
MeasureTheory.Lp.simpleFunc.toSimpleFunc_toLp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ ↑(toSimpleFunc (toLp f hfi)) =ᵐ[μ] ↑f rw [← AEEqFun.mk_eq_mk] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ AEEqFun.mk ↑(toSimpleFunc (toLp f hfi)) ?m.1694496 = AEEqFun.mk ↑f ?m.1694497 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ AEStronglyMeasurable (↑(toSimpleFunc (toLp f hfi))) μ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ AEStronglyMeasurable (↑f) μ exact Classical.choose_spec (toLp f hfi).2 ** Qed
MeasureTheory.Lp.simpleFunc.zero_toSimpleFunc α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α ⊢ ↑(toSimpleFunc 0) =ᵐ[μ] 0 filter_upwards [toSimpleFunc_eq_toFun (0 : Lp.simpleFunc E p μ), Lp.coeFn_zero E 1 μ] with _ h₁ _ case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α a✝¹ : α h₁ : ↑(toSimpleFunc 0) a✝¹ = ↑↑↑0 a✝¹ a✝ : ↑↑0 a✝¹ = OfNat.ofNat 0 a✝¹ ⊢ ↑(toSimpleFunc 0) a✝¹ = OfNat.ofNat 0 a✝¹ rwa [h₁] ** Qed
MeasureTheory.Lp.simpleFunc.add_toSimpleFunc α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } ⊢ ↑(toSimpleFunc (f + g)) =ᵐ[μ] ↑(toSimpleFunc f) + ↑(toSimpleFunc g) filter_upwards [toSimpleFunc_eq_toFun (f + g), toSimpleFunc_eq_toFun f, toSimpleFunc_eq_toFun g, Lp.coeFn_add (f : Lp E p μ) g] with _ case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑↑↑(f + g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑↑(↑f + ↑g) a✝ = (↑↑↑f + ↑↑↑g) a✝ → ↑(toSimpleFunc (f + g)) a✝ = (↑(toSimpleFunc f) + ↑(toSimpleFunc g)) a✝ simp only [AddSubgroup.coe_add, Pi.add_apply] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(toSimpleFunc (f + g)) a✝ = ↑(toSimpleFunc f) a✝ + ↑(toSimpleFunc g) a✝ iterate 4 intro h; rw [h] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝¹ : ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ h✝ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ ⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ intro h case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝² : ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ h✝¹ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h✝ : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ h : ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ ⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ rw [h] ** Qed
MeasureTheory.Lp.simpleFunc.sub_toSimpleFunc α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } ⊢ ↑(toSimpleFunc (f - g)) =ᵐ[μ] ↑(toSimpleFunc f) - ↑(toSimpleFunc g) filter_upwards [toSimpleFunc_eq_toFun (f - g), toSimpleFunc_eq_toFun f, toSimpleFunc_eq_toFun g, Lp.coeFn_sub (f : Lp E p μ) g] with _ case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f - g)) a✝ = ↑↑↑(f - g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑↑(↑f - ↑g) a✝ = (↑↑↑f - ↑↑↑g) a✝ → ↑(toSimpleFunc (f - g)) a✝ = (↑(toSimpleFunc f) - ↑(toSimpleFunc g)) a✝ simp only [AddSubgroup.coe_sub, Pi.sub_apply] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(toSimpleFunc (f - g)) a✝ = ↑(toSimpleFunc f) a✝ - ↑(toSimpleFunc g) a✝ repeat' intro h; rw [h] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝¹ : ↑(toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝ h✝ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ ⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ intro h case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝² : ↑(toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝ h✝¹ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h✝ : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ h : ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ ⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ rw [h] ** Qed
MeasureTheory.Lp.simpleFunc.induction α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } ⊢ P f suffices ∀ f : α →ₛ E, ∀ hf : Memℒp f p μ, P (toLp f hf) by rw [← toLp_toSimpleFunc f] apply this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } ⊢ ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) clear f α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) ⊢ ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) apply SimpleFunc.induction α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } this : ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) ⊢ P f rw [← toLp_toSimpleFunc f] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } this : ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) ⊢ P (toLp (toSimpleFunc f) (_ : Memℒp (↑(toSimpleFunc f)) p)) apply this case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) ⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p), P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) intro c s hs hf case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p ⊢ P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) by_cases hc : c = 0 case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : ¬c = 0 ⊢ P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) exact h_ind c hs (SimpleFunc.measure_lt_top_of_memℒp_indicator hp_pos hp_ne_top hc hs hf) case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) convert h_ind 0 MeasurableSet.empty (by simp) using 1 case h.e'_1 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf = indicatorConst p (_ : MeasurableSet ∅) (_ : ↑↑μ ∅ ≠ ⊤) 0 ext1 case h.e'_1.a α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ ↑(toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) = ↑(indicatorConst p (_ : MeasurableSet ∅) (_ : ↑↑μ ∅ ≠ ⊤) 0) simp [hc] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ ↑↑μ ∅ < ⊤ simp case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) ⊢ ∀ ⦃f g : α →ₛ E⦄, Disjoint (support ↑f) (support ↑g) → (∀ (hf : Memℒp (↑f) p), P (toLp f hf)) → (∀ (hf : Memℒp (↑g) p), P (toLp g hf)) → ∀ (hf : Memℒp (↑(f + g)) p), P (toLp (f + g) hf) intro f g hfg hf hg hfg' case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : ∀ (hf : Memℒp (↑f) p), P (toLp f hf) hg : ∀ (hf : Memℒp (↑g) p), P (toLp g hf) hfg' : Memℒp (↑(f + g)) p ⊢ P (toLp (f + g) hfg') obtain ⟨hf', hg'⟩ : Memℒp f p μ ∧ Memℒp g p μ := (memℒp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable).mp hfg' case h_add.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : ∀ (hf : Memℒp (↑f) p), P (toLp f hf) hg : ∀ (hf : Memℒp (↑g) p), P (toLp g hf) hfg' : Memℒp (↑(f + g)) p hf' : Memℒp (↑f) p hg' : Memℒp (↑g) p ⊢ P (toLp (f + g) hfg') exact h_add hf' hg' hfg (hf hf') (hg hg') ** Qed
MeasureTheory.Lp.simpleFunc.coeFn_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G f g : { x // x ∈ simpleFunc G p μ } ⊢ ↑↑↑f ≤ᵐ[μ] ↑↑↑g ↔ f ≤ g rw [← Subtype.coe_le_coe, ← Lp.coeFn_le] ** Qed
MeasureTheory.Lp.simpleFunc.exists_simpleFunc_nonneg_ae_eq α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G f : { x // x ∈ simpleFunc G p μ } hf : 0 ≤ f ⊢ ∃ f', 0 ≤ f' ∧ ↑↑↑f =ᵐ[μ] ↑f' rcases f with ⟨⟨f, hp⟩, g, (rfl : _ = f)⟩ case mk.mk.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G g : α →ₛ G hp : AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ) ∈ Lp G p hf : 0 ≤ { val := { val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }, property := (_ : ∃ s, AEEqFun.mk ↑s (_ : AEStronglyMeasurable (↑s) μ) = ↑{ val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }) } ⊢ ∃ f', 0 ≤ f' ∧ ↑↑↑{ val := { val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }, property := (_ : ∃ s, AEEqFun.mk ↑s (_ : AEStronglyMeasurable (↑s) μ) = ↑{ val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }) } =ᵐ[μ] ↑f' change 0 ≤ᵐ[μ] g at hf case mk.mk.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G g : α →ₛ G hp : AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ) ∈ Lp G p hf : 0 ≤ᵐ[μ] ↑g ⊢ ∃ f', 0 ≤ f' ∧ ↑↑↑{ val := { val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }, property := (_ : ∃ s, AEEqFun.mk ↑s (_ : AEStronglyMeasurable (↑s) μ) = ↑{ val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }) } =ᵐ[μ] ↑f' refine ⟨g ⊔ 0, le_sup_right, (AEEqFun.coeFn_mk _ _).trans ?_⟩ case mk.mk.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G g : α →ₛ G hp : AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ) ∈ Lp G p hf : 0 ≤ᵐ[μ] ↑g ⊢ ↑g =ᵐ[μ] ↑(g ⊔ 0) exact hf.mono fun x hx ↦ (sup_of_le_left hx).symm ** Qed
MeasureTheory.Memℒp.induction α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f ⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f have : ∀ f : Lp.simpleFunc E p μ, P f := by intro f exact h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memℒp f) (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memℒp f)) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this✝ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f this : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f ⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f have : ∀ f : Lp E p μ, P f := fun f => (Lp.simpleFunc.denseRange hp_ne_top).induction_on f h_closed this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this✝¹ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f this✝ : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f this : ∀ (f : { x // x ∈ Lp E p }), P ↑↑f ⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f exact fun f hf => h_ae hf.coeFn_toLp (Lp.memℒp _) (this (hf.toLp f)) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g ⊢ ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f apply SimpleFunc.induction case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g ⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s), Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p → P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) intro c s hs h case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) by_cases hc : c = 0 case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : ¬c = 0 ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) have hp_pos : p ≠ 0 := (lt_of_lt_of_le zero_lt_one _i.elim).ne' case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : ¬c = 0 hp_pos : p ≠ 0 ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) exact h_ind c hs (SimpleFunc.measure_lt_top_of_memℒp_indicator hp_pos hp_ne_top hc hs h) case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) subst hc case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) convert h_ind 0 MeasurableSet.empty (by simp) using 1 case h.e'_1 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p ⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) = Set.indicator ∅ fun x => 0 ext case h.e'_1.h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p x✝ : α ⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = Set.indicator ∅ (fun x => 0) x✝ simp [const] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p ⊢ ↑↑μ ∅ < ⊤ simp case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g ⊢ ∀ ⦃f g : α →ₛ E⦄, Disjoint (support ↑f) (support ↑g) → (Memℒp (↑f) p → P ↑f) → (Memℒp (↑g) p → P ↑g) → Memℒp (↑(f + g)) p → P ↑(f + g) intro f g hfg hf hg int_fg case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : Memℒp (↑f) p → P ↑f hg : Memℒp (↑g) p → P ↑g int_fg : Memℒp (↑(f + g)) p ⊢ P ↑(f + g) rw [SimpleFunc.coe_add, memℒp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable] at int_fg case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : Memℒp (↑f) p → P ↑f hg : Memℒp (↑g) p → P ↑g int_fg : Memℒp (↑f) p ∧ Memℒp (↑g) p ⊢ P ↑(f + g) refine' h_add hfg int_fg.1 int_fg.2 (hf int_fg.1) (hg int_fg.2) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f ⊢ ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f intro f α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f f : { x // x ∈ Lp.simpleFunc E p μ } ⊢ P ↑↑↑f exact h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memℒp f) (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memℒp f)) ** Qed
MeasureTheory.L1.SimpleFunc.integrable α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α f : { x // x ∈ Lp.simpleFunc E 1 μ } ⊢ Integrable ↑(Lp.simpleFunc.toSimpleFunc f) rw [← memℒp_one_iff_integrable] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α f : { x // x ∈ Lp.simpleFunc E 1 μ } ⊢ Memℒp (↑(Lp.simpleFunc.toSimpleFunc f)) 1 exact Lp.simpleFunc.memℒp f ** Qed
MeasureTheory.analyticSet_empty α : Type u_1 ι : Type u_2 inst✝ : TopologicalSpace α ⊢ AnalyticSet ∅ rw [AnalyticSet] α : Type u_1 ι : Type u_2 inst✝ : TopologicalSpace α ⊢ ∅ = ∅ ∨ ∃ f, Continuous f ∧ range f = ∅ exact Or.inl rfl ** Qed
IsOpen.analyticSet_image α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α β : Type u_3 inst✝¹ : TopologicalSpace β inst✝ : PolishSpace β s : Set β hs : IsOpen s f : β → α f_cont : Continuous f ⊢ AnalyticSet (f '' s) rw [image_eq_range] α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α β : Type u_3 inst✝¹ : TopologicalSpace β inst✝ : PolishSpace β s : Set β hs : IsOpen s f : β → α f_cont : Continuous f ⊢ AnalyticSet (range fun x => f ↑x) haveI : PolishSpace s := hs.polishSpace α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α β : Type u_3 inst✝¹ : TopologicalSpace β inst✝ : PolishSpace β s : Set β hs : IsOpen s f : β → α f_cont : Continuous f this : PolishSpace ↑s ⊢ AnalyticSet (range fun x => f ↑x) exact analyticSet_range_of_polishSpace (f_cont.comp continuous_subtype_val) ** Qed
MeasureTheory.AnalyticSet.image_of_continuousOn α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s ⊢ AnalyticSet (f '' s) rcases analyticSet_iff_exists_polishSpace_range.1 hs with ⟨γ, γtop, γpolish, g, g_cont, gs⟩ case intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s ⊢ AnalyticSet (f '' s) have : f '' s = range (f ∘ g) := by rw [range_comp, gs] case intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ AnalyticSet (f '' s) rw [this] case intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ AnalyticSet (range (f ∘ g)) apply analyticSet_range_of_polishSpace case intro.intro.intro.intro.intro.f_cont α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ Continuous (f ∘ g) apply hf.comp_continuous g_cont fun x => _ α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ ∀ (x : γ), g x ∈ s rw [← gs] α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ ∀ (x : γ), g x ∈ range g exact mem_range_self α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s ⊢ f '' s = range (f ∘ g) rw [range_comp, gs] ** Qed
MeasureTheory.AnalyticSet.iInter α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α hι : Nonempty ι inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) ⊢ AnalyticSet (⋂ n, s n) rcases hι with ⟨i₀⟩ case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι ⊢ AnalyticSet (⋂ n, s n) choose β hβ h'β f f_cont f_range using fun n => analyticSet_iff_exists_polishSpace_range.1 (hs n) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n ⊢ AnalyticSet (⋂ n, s n) let γ := ∀ n, β n case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n ⊢ AnalyticSet (⋂ n, s n) let t : Set γ := ⋂ n, { x \| f n (x n) = f i₀ (x i₀) } case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} ⊢ AnalyticSet (⋂ n, s n) have t_closed : IsClosed t := by apply isClosed_iInter intro n exact isClosed_eq ((f_cont n).comp (continuous_apply n)) ((f_cont i₀).comp (continuous_apply i₀)) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t ⊢ AnalyticSet (⋂ n, s n) haveI : PolishSpace t := t_closed.polishSpace case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t ⊢ AnalyticSet (⋂ n, s n) let F : t → α := fun x => f i₀ ((x : γ) i₀) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) ⊢ AnalyticSet (⋂ n, s n) have F_cont : Continuous F := (f_cont i₀).comp ((continuous_apply i₀).comp continuous_subtype_val) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F F_range : range F = ⋂ n, s n ⊢ AnalyticSet (⋂ n, s n) rw [← F_range] case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F F_range : range F = ⋂ n, s n ⊢ AnalyticSet (range F) exact analyticSet_range_of_polishSpace F_cont α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} ⊢ IsClosed t apply isClosed_iInter case h α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} ⊢ ∀ (i : ι), IsClosed {x \| f i (x i) = f i₀ (x i₀)} intro n case h α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} n : ι ⊢ IsClosed {x \| f n (x n) = f i₀ (x i₀)} exact isClosed_eq ((f_cont n).comp (continuous_apply n)) ((f_cont i₀).comp (continuous_apply i₀)) α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F ⊢ range F = ⋂ n, s n apply Subset.antisymm case h₁ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F ⊢ range F ⊆ ⋂ n, s n rintro y ⟨x, rfl⟩ case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t ⊢ F x ∈ ⋂ n, s n refine mem_iInter.2 fun n => ?_ case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t n : ι ⊢ F x ∈ s n have : f n ((x : γ) n) = F x := (mem_iInter.1 x.2 n : _) case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this✝ : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t n : ι this : f n (↑x n) = F x ⊢ F x ∈ s n rw [← this, ← f_range n] case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this✝ : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t n : ι this : f n (↑x n) = F x ⊢ f n (↑x n) ∈ range (f n) exact mem_range_self _ case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F ⊢ ⋂ n, s n ⊆ range F intro y hy case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n ⊢ y ∈ range F have A : ∀ n, ∃ x : β n, f n x = y := by intro n rw [← mem_range, f_range n] exact mem_iInter.1 hy n case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n A : ∀ (n : ι), ∃ x, f n x = y ⊢ y ∈ range F choose x hx using A case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y ⊢ y ∈ range F have xt : x ∈ t := by refine mem_iInter.2 fun n => ?_ simp [hx] case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y xt : x ∈ t ⊢ y ∈ range F refine' ⟨⟨x, xt⟩, _⟩ case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y xt : x ∈ t ⊢ F { val := x, property := xt } = y exact hx i₀ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n ⊢ ∀ (n : ι), ∃ x, f n x = y intro n α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n n : ι ⊢ ∃ x, f n x = y rw [← mem_range, f_range n] α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n n : ι ⊢ y ∈ s n exact mem_iInter.1 hy n α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y ⊢ x ∈ t refine mem_iInter.2 fun n => ?_ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y n : ι ⊢ x ∈ {x \| f n (x n) = f i₀ (x i₀)} simp [hx] ** Qed
IsClosed.analyticSet α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : PolishSpace α s : Set α hs : IsClosed s ⊢ AnalyticSet s haveI : PolishSpace s := hs.polishSpace α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : PolishSpace α s : Set α hs : IsClosed s this : PolishSpace ↑s ⊢ AnalyticSet s rw [← @Subtype.range_val α s] α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : PolishSpace α s : Set α hs : IsClosed s this : PolishSpace ↑s ⊢ AnalyticSet (range Subtype.val) exact analyticSet_range_of_polishSpace continuous_subtype_val ** Qed
MeasurableSet.analyticSet_image α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f ⊢ AnalyticSet (f '' s) letI := upgradeStandardBorel X α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X ⊢ AnalyticSet (f '' s) rcases hf.exists_continuous with ⟨τ', hle, hfc, hτ'⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X ⊢ AnalyticSet (f '' s) letI m' : MeasurableSpace X := @borel _ τ' case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X m' : MeasurableSpace X := borel X ⊢ AnalyticSet (f '' s) haveI b' : BorelSpace X := ⟨rfl⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X m' : MeasurableSpace X := borel X b' : BorelSpace X ⊢ AnalyticSet (f '' s) have hle := borel_anti hle case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle✝ : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X m' : MeasurableSpace X := borel X b' : BorelSpace X hle : borel X ≤ borel X ⊢ AnalyticSet (f '' s) exact (hle _ hs).analyticSet.image_of_continuous hfc ** Qed
MeasureTheory.AnalyticSet.measurableSet_of_compl α : Type u_1 ι : Type u_2 inst✝³ : TopologicalSpace α inst✝² : T2Space α inst✝¹ : MeasurableSpace α inst✝ : OpensMeasurableSpace α s : Set α hs : AnalyticSet s hsc : AnalyticSet sᶜ ⊢ MeasurableSet s rcases hs.measurablySeparable hsc disjoint_compl_right with ⟨u, hsu, hdu, hmu⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝³ : TopologicalSpace α inst✝² : T2Space α inst✝¹ : MeasurableSpace α inst✝ : OpensMeasurableSpace α s : Set α hs : AnalyticSet s hsc : AnalyticSet sᶜ u : Set α hsu : s ⊆ u hdu : Disjoint sᶜ u hmu : MeasurableSet u ⊢ MeasurableSet s obtain rfl : s = u := hsu.antisymm (disjoint_compl_left_iff_subset.1 hdu) case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝³ : TopologicalSpace α inst✝² : T2Space α inst✝¹ : MeasurableSpace α inst✝ : OpensMeasurableSpace α s : Set α hs : AnalyticSet s hsc : AnalyticSet sᶜ hsu : s ⊆ s hdu : Disjoint sᶜ s hmu : MeasurableSet s ⊢ MeasurableSet s exact hmu ** Qed
Measurable.map_measurableSpace_eq_borel α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f ⊢ MeasurableSpace.map f inst✝⁷ = borel Y have d := hf.mono le_rfl OpensMeasurableSpace.borel_le α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f d : Measurable f ⊢ MeasurableSpace.map f inst✝⁷ = borel Y letI := borel Y α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f d : Measurable f this : MeasurableSpace Y := borel Y ⊢ MeasurableSpace.map f inst✝⁷ = borel Y haveI : BorelSpace Y := ⟨rfl⟩ α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f d : Measurable f this✝ : MeasurableSpace Y := borel Y this : BorelSpace Y ⊢ MeasurableSpace.map f inst✝⁷ = borel Y exact d.map_measurableSpace_eq hsurj ** Qed
Measurable.measurableSet_preimage_iff_inter_range α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β f : X → Y inst✝ : SecondCountableTopology ↑(range f) hf : Measurable f hr : MeasurableSet (range f) s : Set Y ⊢ MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f) rw [hf.measurableSet_preimage_iff_preimage_val, ← (MeasurableEmbedding.subtype_coe hr).measurableSet_image, Subtype.image_preimage_coe] ** Qed
Continuous.map_eq_borel α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : PolishSpace X inst✝⁴ : MeasurableSpace X inst✝³ : BorelSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f ⊢ MeasurableSpace.map f inst✝⁴ = borel Y borelize Y α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : PolishSpace X inst✝⁴ : MeasurableSpace X inst✝³ : BorelSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f this✝¹ : MeasurableSpace Y := borel Y this✝ : BorelSpace Y ⊢ MeasurableSpace.map f inst✝⁴ = borel Y exact hf.measurable.map_measurableSpace_eq hsurj ** Qed
Continuous.map_borel_eq α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁴ : TopologicalSpace X inst✝³ : PolishSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f ⊢ MeasurableSpace.map f (borel X) = borel Y borelize X α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁴ : TopologicalSpace X inst✝³ : PolishSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f this✝¹ : MeasurableSpace X := borel X this✝ : BorelSpace X ⊢ MeasurableSpace.map f (borel X) = borel Y exact hf.map_eq_borel hsurj ** Qed
ContinuousOn.measurableEmbedding α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s ⊢ ∀ ⦃s_1 : Set ↑s⦄, MeasurableSet s_1 → MeasurableSet (restrict s f '' s_1) intro u hu α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s u : Set ↑s hu : MeasurableSet u ⊢ MeasurableSet (restrict s f '' u) have A : MeasurableSet (((↑) : s → γ) '' u) := (MeasurableEmbedding.subtype_coe hs).measurableSet_image.2 hu α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s u : Set ↑s hu : MeasurableSet u A : MeasurableSet (Subtype.val '' u) ⊢ MeasurableSet (restrict s f '' u) have B : MeasurableSet (f '' (((↑) : s → γ) '' u)) := A.image_of_continuousOn_injOn (f_cont.mono (Subtype.coe_image_subset s u)) (f_inj.mono (Subtype.coe_image_subset s u)) α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s u : Set ↑s hu : MeasurableSet u A : MeasurableSet (Subtype.val '' u) B : MeasurableSet (f '' (Subtype.val '' u)) ⊢ MeasurableSet (restrict s f '' u) rwa [← image_comp] at B ** Qed
MeasureTheory.isClopenable_iff_measurableSet α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁴ : T2Space β inst✝³ : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ ⊢ IsClopenable s ↔ MeasurableSet s refine' ⟨fun hs => _, fun hs => hs.isClopenable⟩ α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁴ : T2Space β inst✝³ : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : IsClopenable s ⊢ MeasurableSet s borelize γ α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ hs : IsClopenable s inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ ⊢ MeasurableSet s obtain ⟨t', t't, t'_polish, _, s_open⟩ : ∃ t' : TopologicalSpace γ, t' ≤ tγ ∧ @PolishSpace γ t' ∧ IsClosed[t'] s ∧ IsOpen[t'] s := hs case intro.intro.intro.intro α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ t' : TopologicalSpace γ t't : t' ≤ tγ t'_polish : PolishSpace γ left✝ : IsClosed s s_open : IsOpen s ⊢ MeasurableSet s rw [← borel_eq_borel_of_le t'_polish _ t't] α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ t' : TopologicalSpace γ t't : t' ≤ tγ t'_polish : PolishSpace γ left✝ : IsClosed s s_open : IsOpen s ⊢ PolishSpace γ infer_instance case intro.intro.intro.intro α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ t' : TopologicalSpace γ t't : t' ≤ tγ t'_polish : PolishSpace γ left✝ : IsClosed s s_open : IsOpen s ⊢ MeasurableSet s exact MeasurableSpace.measurableSet_generateFrom s_open ** Qed
MeasurableSet.isClopenable' α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s ⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s letI := upgradeStandardBorel α α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α ⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s obtain ⟨t, hle, ht, s_clopen⟩ := hs.isClopenable case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s refine' ⟨t, _, ht, s_clopen⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ BorelSpace α constructor case intro.intro.intro.measurable_eq α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ inst✝¹ = borel α rw [eq_borel_upgradeStandardBorel α, borel_eq_borel_of_le ht _ hle] α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ PolishSpace α infer_instance ** Qed
MeasurableSet.standardBorel α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s ⊢ StandardBorelSpace ↑s obtain ⟨_, _, _, s_closed, _⟩ := hs.isClopenable' case intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s w✝ : TopologicalSpace α left✝¹ : BorelSpace α left✝ : PolishSpace α s_closed : IsClosed s right✝ : IsOpen s ⊢ StandardBorelSpace ↑s haveI := s_closed.polishSpace case intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s w✝ : TopologicalSpace α left✝¹ : BorelSpace α left✝ : PolishSpace α s_closed : IsClosed s right✝ : IsOpen s this : PolishSpace ↑s ⊢ StandardBorelSpace ↑s infer_instance ** Qed
MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite α : Type u_1 ι : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : StandardBorelSpace α inst✝ : Finite α ⊢ ∃ n, Nonempty (α ≃ᵐ ↑(range fun x => ↑↑x)) obtain ⟨n, ⟨n_equiv⟩⟩ := Finite.exists_equiv_fin α case intro.intro α : Type u_1 ι : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : StandardBorelSpace α inst✝ : Finite α n : ℕ n_equiv : α ≃ Fin n ⊢ ∃ n, Nonempty (α ≃ᵐ ↑(range fun x => ↑↑x)) refine' ⟨n, ⟨PolishSpace.Equiv.measurableEquiv (n_equiv.trans _)⟩⟩ case intro.intro α : Type u_1 ι : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : StandardBorelSpace α inst✝ : Finite α n : ℕ n_equiv : α ≃ Fin n ⊢ Fin n ≃ ↑(range fun x => ↑↑x) exact Equiv.ofInjective _ (Nat.cast_injective.comp Fin.val_injective) ** Qed
MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α ⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast)) have : PolishSpace (range ((↑) : ℕ → ℝ)) := Nat.closedEmbedding_coe_real.isClosedMap.closed_range.polishSpace α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α this : PolishSpace ↑(range Nat.cast) ⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast)) refine' ⟨PolishSpace.Equiv.measurableEquiv _⟩ α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α this : PolishSpace ↑(range Nat.cast) ⊢ α ≃ ↑(range Nat.cast) refine' (nonempty_equiv_of_countable.some : α ≃ ℕ).trans _ α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α this : PolishSpace ↑(range Nat.cast) ⊢ ℕ ≃ ↑(range Nat.cast) exact Equiv.ofInjective ((↑) : ℕ → ℝ) Nat.cast_injective ** Qed
Besicovitch.SatelliteConfig.inter' α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) rcases lt_or_le i (last N) with (H \| H) case inl α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : i < last N ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) exact a.inter i H case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : last N ≤ i ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) have I : i = last N := top_le_iff.1 H case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : last N ≤ i I : i = last N ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) have := (a.rpos (last N)).le case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : last N ≤ i I : i = last N this : 0 ≤ r a (last N) ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) simp only [I, add_nonneg this this, dist_self] ** Qed
Besicovitch.SatelliteConfig.hlast' ** α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ ⊢ r a (last N) ≤ τ * r a i rcases lt_or_le i (last N) with (H \| H) case inl α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : i < last N ⊢ r a (last N) ≤ τ * r a i exact (a.hlast i H).2 case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : last N ≤ i ⊢ r a (last N) ≤ τ * r a i have : i = last N := top_le_iff.1 H case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : last N ≤ i this : i = last N ⊢ r a (last N) ≤ τ * r a i rw [this] case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : last N ≤ i this : i = last N ⊢ r a (last N) ≤ τ * r a (last N) exact le_mul_of_one_le_left (a.rpos _).le h Qed
Besicovitch.TauPackage.monotone_iUnionUpTo α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α ⊢ Monotone (iUnionUpTo p) intro i j hij α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i j : Ordinal.{u} hij : i ≤ j ⊢ iUnionUpTo p i ≤ iUnionUpTo p j simp only [iUnionUpTo] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i j : Ordinal.{u} hij : i ≤ j ⊢ ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ≤ ⋃ j_1, ball (BallPackage.c p.toBallPackage (index p ↑j_1)) (BallPackage.r p.toBallPackage (index p ↑j_1)) exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩ ** Qed
Besicovitch.TauPackage.lastStep_nonempty ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α ⊢ Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} by_contra h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ False suffices H : Function.Injective p.index α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} H : Function.Injective (index p) ⊢ False case H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ Function.Injective (index p) exact not_injective_of_ordinal p.index H case H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ Function.Injective (index p) intro x y hxy case H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y ⊢ x = y wlog x_le_y : x ≤ y generalizing x y α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y ⊢ x = y rcases eq_or_lt_of_le x_le_y with (rfl \| H) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y ⊢ x = y simp only [nonempty_def, not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] at h case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∀ (x : Ordinal.{u}), ∃ x_1, ¬BallPackage.c p.toBallPackage x_1 ∈ iUnionUpTo p x ∧ R p x ≤ p.τ * BallPackage.r p.toBallPackage x_1 ⊢ x = y specialize h y case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ x = y ** have A : p.c (p.index y) ∉ p.iUnionUpTo y := by have : p.index y = Classical.epsilon fun b : β => p.c b ∉ p.iUnionUpTo y ∧ p.R y ≤ p.τ * p.r b := by rw [TauPackage.index]; rfl rw [this] exact (Classical.epsilon_spec h).1 ** case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : ¬BallPackage.c p.toBallPackage (index p y) ∈ iUnionUpTo p y ⊢ x = y simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, Subtype.exists, Subtype.coe_mk] at A case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : ∀ (x : Ordinal.{u}), x < y → ¬BallPackage.c p.toBallPackage (index p y) ∈ ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ⊢ x = y specialize A x H case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : ¬BallPackage.c p.toBallPackage (index p y) ∈ ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ⊢ x = y simp [hxy] at A case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : BallPackage.r p.toBallPackage (index p y) ≤ 0 ⊢ x = y exact (lt_irrefl _ ((p.rpos (p.index y)).trans_le A)).elim case H.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y this : ∀ ⦃x y : Ordinal.{u}⦄, index p x = index p y → x ≤ y → x = y x_le_y : ¬x ≤ y ⊢ x = y exact (this hxy.symm (le_of_not_le x_le_y)).symm case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x : Ordinal.{u} hxy : index p x = index p x x_le_y : x ≤ x ⊢ x = x rfl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ ¬BallPackage.c p.toBallPackage (index p y) ∈ iUnionUpTo p y ** have : p.index y = Classical.epsilon fun b : β => p.c b ∉ p.iUnionUpTo y ∧ p.R y ≤ p.τ * p.r b := by rw [TauPackage.index]; rfl ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x this : index p y = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b ⊢ ¬BallPackage.c p.toBallPackage (index p y) ∈ iUnionUpTo p y rw [this] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x this : index p y = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b ⊢ ¬BallPackage.c p.toBallPackage (Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b) ∈ iUnionUpTo p y exact (Classical.epsilon_spec h).1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ index p y = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b rw [TauPackage.index] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ (Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∧ ⨆ b, BallPackage.r p.toBallPackage ↑b ≤ p.τ * BallPackage.r p.toBallPackage b) = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b rfl Qed
Besicovitch.TauPackage.mem_iUnionUpTo_lastStep α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β ⊢ BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) have A : ∀ z : β, p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep := by have : p.lastStep ∈ {i \| ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} := csInf_mem p.lastStep_nonempty simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem] ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) ⊢ BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) by_contra h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) ⊢ False rcases A x with (H \| H) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) ⊢ False have Rpos : 0 < p.R p.lastStep := by apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) ⊢ False ** have B : p.τ⁻¹ * p.R p.lastStep < p.R p.lastStep := by conv_rhs => rw [← one_mul (p.R p.lastStep)] exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one ** case inr.intro.intro α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y ⊢ False rcases A y with (Hy \| Hy) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β ⊢ ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) ** have : p.lastStep ∈ {i \| ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} := csInf_mem p.lastStep_nonempty ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β this : lastStep p ∈ {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem] case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) ⊢ False exact h H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) ⊢ 0 < R p (lastStep p) apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) ⊢ p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) conv_rhs => rw [← one_mul (p.R p.lastStep)] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) ⊢ p.τ⁻¹ * R p (lastStep p) < 1 * R p (lastStep p) exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) ⊢ ∃ y, ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) ∧ p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y have := exists_lt_of_lt_csSup ?_ B case refine_1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) ⊢ Set.Nonempty (range fun b => BallPackage.r p.toBallPackage ↑b) rw [← image_univ, nonempty_image_iff] case refine_1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) ⊢ Set.Nonempty univ exact ⟨⟨_, h⟩, mem_univ _⟩ case refine_2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) this : ∃ a, (a ∈ range fun b => BallPackage.r p.toBallPackage ↑b) ∧ p.τ⁻¹ * R p (lastStep p) < a ⊢ ∃ y, ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) ∧ p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists, Subtype.coe_mk] case inr.intro.intro.inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y Hy : BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) ⊢ False exact hy1 Hy case inr.intro.intro.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p) ⊢ False rw [← div_eq_inv_mul] at hy2 case inr.intro.intro.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : R p (lastStep p) / p.τ < BallPackage.r p.toBallPackage y Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p) ⊢ False have := (div_le_iff' (_root_.zero_lt_one.trans p.one_lt_tau)).1 hy2.le case inr.intro.intro.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : R p (lastStep p) / p.τ < BallPackage.r p.toBallPackage y Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p) this : R p (lastStep p) ≤ p.τ * BallPackage.r p.toBallPackage y ⊢ False exact lt_irrefl _ (Hy.trans_le this) Qed
Besicovitch.TauPackage.color_lt α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i : Ordinal.{u} hi : i < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) ⊢ color p i < N induction' i using Ordinal.induction with i IH case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p ⊢ color p i < N let A : Set ℕ := ⋃ (j : { j // j < i }) (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty), {p.color j} case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} ⊢ color p i < N have color_i : p.color i = sInf (univ \ A) := by rw [color] case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ color p i < N rw [color_i] case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ sInf (univ \ A) < N have N_mem : N ∈ univ \ A := by simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] intro j ji _ exact (IH j ji (ji.trans hi)).ne' case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A ⊢ sInf (univ \ A) ≠ N intro Inf_eq_N case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N ⊢ False have : ∀ k, k < N → ∃ j, j < i ∧ (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧ k = p.color j := by intro k hk rw [← Inf_eq_N] at hk have : k ∈ A := by simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk simp [and_assoc, -exists_and_left] at this simpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists, Subtype.coe_mk] case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N this : ∀ (k : ℕ), k < N → ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j ⊢ False choose! g hg using this case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) ⊢ False let G : ℕ → Ordinal := fun n => if n = N then i else g n case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p ⊢ False have fGn : ∀ n, n ≤ N → p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n)) := by intro n hn have : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by rw [index]; rfl rw [this] have : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _) exact Classical.epsilon_spec this ** case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) sc : SatelliteConfig α N p.τ := { c := fun k => BallPackage.c p.toBallPackage (index p (G ↑k)), r := fun k => BallPackage.r p.toBallPackage (index p (G ↑k)), rpos := (_ : ∀ (k : Fin (Nat.succ N)), 0 < BallPackage.r p.toBallPackage (index p (G ↑k))), h := (_ : ∀ (a b : Fin (Nat.succ N)), a ≠ b → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b), hlast := (_ : ∀ (a : Fin (N + 1)), a < last N → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑(last N)))) ∧ BallPackage.r p.toBallPackage (index p (G ↑(last N))) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a))), inter := (_ : ∀ (a : Fin (N + 1)), a < last N → dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N)) } ⊢ False exact hN.false sc α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} ⊢ color p i = sInf (univ \ A) rw [color] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ N ∈ univ \ A simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ ∀ (x : Ordinal.{u}), x < i → Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) → ¬N = color p x intro j ji _ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) j : Ordinal.{u} ji : j < i a✝ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ⊢ ¬N = color p j exact (IH j ji (ji.trans hi)).ne' α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A this : sInf (univ \ A) ≠ N ⊢ sInf (univ \ A) < N rcases (csInf_le (OrderBot.bddBelow (univ \ A)) N_mem).lt_or_eq with (H \| H) case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A this : sInf (univ \ A) ≠ N H : sInf (univ \ A) < N ⊢ sInf (univ \ A) < N exact H case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A this : sInf (univ \ A) ≠ N H : sInf (univ \ A) = N ⊢ sInf (univ \ A) < N exact (this H).elim α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N ⊢ ∀ (k : ℕ), k < N → ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j intro k hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < N ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j rw [← Inf_eq_N] at hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j have : k ∈ A := by simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) this : k ∈ A ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j simp [and_assoc, -exists_and_left] at this α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) this : ∃ a, a < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p a)) (BallPackage.r p.toBallPackage (index p a)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p a ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j simpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists, Subtype.coe_mk] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) ⊢ k ∈ A simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n ⊢ ∀ (n : ℕ), n ≤ N → color p (G n) = n intro n hn α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n n : ℕ hn : n ≤ N ⊢ color p (G n) = n rcases hn.eq_or_lt with (rfl \| H) case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 hn : n ≤ n ⊢ color p (G n) = n simp only case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 hn : n ≤ n ⊢ color p (if True then i else g n) = n simp only [color_i, Inf_eq_N, if_true, eq_self_iff_true] case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n n : ℕ hn : n ≤ N H : n < N ⊢ color p (G n) = n simp only case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n n : ℕ hn : n ≤ N H : n < N ⊢ color p (if n = N then i else g n) = n simp only [H.ne, (hg n H).right.right.symm, if_false] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n ⊢ ∀ (n : ℕ), n ≤ N → G n < lastStep p intro n hn α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n n : ℕ hn : n ≤ N ⊢ G n < lastStep p rcases hn.eq_or_lt with (rfl \| H) case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 color_G : ∀ (n_1 : ℕ), n_1 ≤ n → color p (G n_1) = n_1 hn : n ≤ n ⊢ G n < lastStep p simp only case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 color_G : ∀ (n_1 : ℕ), n_1 ≤ n → color p (G n_1) = n_1 hn : n ≤ n ⊢ (if True then i else g n) < lastStep p simp only [hi, if_true, eq_self_iff_true] case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n n : ℕ hn : n ≤ N H : n < N ⊢ G n < lastStep p simp only case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n n : ℕ hn : n ≤ N H : n < N ⊢ (if n = N then i else g n) < lastStep p simp only [H.ne, (hg n H).left.trans hi, if_false] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p ⊢ ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) intro n hn α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N ⊢ ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) ** have : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by rw [index]; rfl ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) rw [this] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ¬BallPackage.c p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) ** have : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _) ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this✝ : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t this : ∃ t, ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ¬BallPackage.c p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) exact Classical.epsilon_spec this α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N ⊢ index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t rw [index] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N ⊢ (Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∧ ⨆ b, BallPackage.r p.toBallPackage ↑b ≤ p.τ * BallPackage.r p.toBallPackage b) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t rfl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ∃ t, ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) ⊢ ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) intro a b G_lt α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) have ha : (a : ℕ) ≤ N := Nat.lt_succ_iff.1 a.2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) have hb : (b : ℕ) ≤ N := Nat.lt_succ_iff.1 b.2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) constructor case left α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) have := (fGn b hb).1 case left α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N this : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑b) ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, Subtype.exists, Subtype.coe_mk] at this case left α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N this : ∀ (x : Ordinal.{u}), (x < if ↑b = N then i else g ↑b) → ¬BallPackage.c p.toBallPackage (index p (if ↑b = N then i else g ↑b)) ∈ ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) simpa only [dist_comm, mem_ball, not_lt] using this (G a) G_lt case right α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) apply le_trans _ (fGn a ha).2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ R p (G ↑a) have B : p.c (p.index (G b)) ∉ p.iUnionUpTo (G a) := by intro H; exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ R p (G ↑a) let b' : { t // p.c t ∉ p.iUnionUpTo (G a) } := ⟨p.index (G b), B⟩ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ R p (G ↑a) apply @le_ciSup _ _ _ (fun t : { t // p.c t ∉ p.iUnionUpTo (G a) } => p.r t) _ b' α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } ⊢ BddAbove (range fun t => BallPackage.r p.toBallPackage ↑t) refine' ⟨p.r_bound, fun t ht => _⟩ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ ht : t ∈ range fun t => BallPackage.r p.toBallPackage ↑t ⊢ t ≤ p.r_bound simp only [exists_prop, mem_range, Subtype.exists, Subtype.coe_mk] at ht α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ ht : ∃ a_1, ¬BallPackage.c p.toBallPackage a_1 ∈ iUnionUpTo p (if ↑a = N then i else g ↑a) ∧ BallPackage.r p.toBallPackage a_1 = t ⊢ t ≤ p.r_bound rcases ht with ⟨u, hu⟩ case intro α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ u : β hu : ¬BallPackage.c p.toBallPackage u ∈ iUnionUpTo p (if ↑a = N then i else g ↑a) ∧ BallPackage.r p.toBallPackage u = t ⊢ t ≤ p.r_bound rw [← hu.2] case intro α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ u : β hu : ¬BallPackage.c p.toBallPackage u ∈ iUnionUpTo p (if ↑a = N then i else g ↑a) ∧ BallPackage.r p.toBallPackage u = t ⊢ BallPackage.r p.toBallPackage u ≤ p.r_bound exact p.r_le _ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) intro H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N H : BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) ⊢ False exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) ⊢ ∀ (i j : Fin (Nat.succ N)), i ≠ j → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) j) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) j ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) j ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) j) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) j intro a b a_ne_b α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b wlog G_le : G a ≤ G b generalizing a b α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b G_lt : G ↑a < G ↑b ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b exact Or.inl (Gab a b G_lt) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b this : ∀ (a b : Fin (Nat.succ N)), a ≠ b → G ↑a ≤ G ↑b → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b G_le : ¬G ↑a ≤ G ↑b ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b exact (this b a a_ne_b.symm (le_of_not_le G_le)).symm α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b ⊢ G ↑a < G ↑b rcases G_le.lt_or_eq with (H \| H) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a = G ↑b ⊢ G ↑a < G ↑b have A : (a : ℕ) ≠ b := Fin.val_injective.ne a_ne_b case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A✝ : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A✝) N_mem : N ∈ univ \ A✝ Inf_eq_N : sInf (univ \ A✝) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a = G ↑b A : ↑a ≠ ↑b ⊢ G ↑a < G ↑b rw [← color_G a (Nat.lt_succ_iff.1 a.2), ← color_G b (Nat.lt_succ_iff.1 b.2), H] at A case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A✝ : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A✝) N_mem : N ∈ univ \ A✝ Inf_eq_N : sInf (univ \ A✝) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a = G ↑b A : color p (G ↑b) ≠ color p (G ↑b) ⊢ G ↑a < G ↑b exact (A rfl).elim case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a < G ↑b ⊢ G ↑a < G ↑b exact H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) ⊢ ∀ (i : Fin (N + 1)), i < last N → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i intro a ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a have I : (a : ℕ) < N := ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a have : G a < G (Fin.last N) := by dsimp; simp [I.ne, (hg a I).1] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N this : G ↑a < G ↑(last N) ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a exact Gab _ _ this α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ G ↑a < G ↑(last N) dsimp α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ (if ↑a = N then i else g ↑a) < if N = N then i else g N simp [I.ne, (hg a I).1] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) ⊢ ∀ (i : Fin (N + 1)), i < last N → dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) intro a ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) have I : (a : ℕ) < N := ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) have J : G (Fin.last N) = i := by dsimp; simp only [if_true, eq_self_iff_true] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) have K : G a = g a := by dsimp; simp [I.ne, (hg a I).1] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i K : G ↑a = g ↑a ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) convert dist_le_add_of_nonempty_closedBall_inter_closedBall (hg _ I).2.1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ G ↑(last N) = i dsimp α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ (if N = N then i else g N) = i simp only [if_true, eq_self_iff_true] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i ⊢ G ↑a = g ↑a dsimp α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i ⊢ (if ↑a = N then i else g ↑a) = g ↑a simp [I.ne, (hg a I).1] Qed
Besicovitch.ae_tendsto_rnDeriv α : Type u_1 inst✝¹¹ : MetricSpace α β : Type u inst✝¹⁰ : SecondCountableTopology α inst✝⁹ : MeasurableSpace α inst✝⁸ : OpensMeasurableSpace α inst✝⁷ : HasBesicovitchCovering α inst✝⁶ : MetricSpace β inst✝⁵ : MeasurableSpace β inst✝⁴ : BorelSpace β inst✝³ : SecondCountableTopology β inst✝² : HasBesicovitchCovering β ρ μ : Measure β inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsLocallyFiniteMeasure ρ ⊢ ∀ᵐ (x : β) ∂μ, Tendsto (fun r => ↑↑ρ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (Measure.rnDeriv ρ μ x)) filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (Besicovitch.vitaliFamily μ) ρ] with x hx case h α : Type u_1 inst✝¹¹ : MetricSpace α β : Type u inst✝¹⁰ : SecondCountableTopology α inst✝⁹ : MeasurableSpace α inst✝⁸ : OpensMeasurableSpace α inst✝⁷ : HasBesicovitchCovering α inst✝⁶ : MetricSpace β inst✝⁵ : MeasurableSpace β inst✝⁴ : BorelSpace β inst✝³ : SecondCountableTopology β inst✝² : HasBesicovitchCovering β ρ μ : Measure β inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsLocallyFiniteMeasure ρ x : β hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x) (𝓝 (Measure.rnDeriv ρ μ x)) ⊢ Tendsto (fun r => ↑↑ρ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (Measure.rnDeriv ρ μ x)) exact hx.comp (tendsto_filterAt μ x) ** Qed
Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β hs : MeasurableSet s ⊢ ∀ᵐ (x : β) ∂μ, Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (indicator s 1 x)) filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet (Besicovitch.vitaliFamily μ) hs] case h α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β hs : MeasurableSet s ⊢ ∀ (a : β), Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) a) (𝓝 (indicator s 1 a)) → Tendsto (fun r => ↑↑μ (s ∩ closedBall a r) / ↑↑μ (closedBall a r)) (𝓝[Ioi 0] 0) (𝓝 (indicator s 1 a)) intro x hx case h α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β hs : MeasurableSet s x : β hx : Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x) (𝓝 (indicator s 1 x)) ⊢ Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (indicator s 1 x)) exact hx.comp (tendsto_filterAt μ x) ** Qed
Besicovitch.ae_tendsto_measure_inter_div α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β ⊢ ∀ᵐ (x : β) ∂Measure.restrict μ s, Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1) filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div (Besicovitch.vitaliFamily μ) s] with x hx using hx.comp (tendsto_filterAt μ x) ** Qed
MeasureTheory.tendstoInMeasure_iff_norm α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : SeminormedAddCommGroup E l : Filter ι f : ι → α → E g : α → E ⊢ TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ), 0 < ε → Tendsto (fun i => ↑↑μ {x \| ε ≤ ‖f i x - g x‖}) l (𝓝 0) simp_rw [TendstoInMeasure, dist_eq_norm] ** Qed
MeasureTheory.TendstoInMeasure.congr' α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ⊢ TendstoInMeasure μ f' l g' intro ε hε α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε ⊢ Tendsto (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) l (𝓝 0) suffices (fun i => μ { x \| ε ≤ dist (f' i x) (g' x) }) =ᶠ[l] fun i => μ { x \| ε ≤ dist (f i x) (g x) } by rw [tendsto_congr' this] exact h_tendsto ε hε α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε ⊢ (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) =ᶠ[l] fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)} filter_upwards [h_left] with i h_ae_eq case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i ⊢ ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)} = ↑↑μ {x \| ε ≤ dist (f i x) (g x)} refine' measure_congr _ case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i ⊢ {x \| ε ≤ dist (f' i x) (g' x)} =ᵐ[μ] {x \| ε ≤ dist (f i x) (g x)} filter_upwards [h_ae_eq, h_right] with x hxf hxg case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i x : α hxf : f i x = f' i x hxg : g x = g' x ⊢ setOf (fun x => ε ≤ dist (f' i x) (g' x)) x = setOf (fun x => ε ≤ dist (f i x) (g x)) x rw [eq_iff_iff] case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i x : α hxf : f i x = f' i x hxg : g x = g' x ⊢ setOf (fun x => ε ≤ dist (f' i x) (g' x)) x ↔ setOf (fun x => ε ≤ dist (f i x) (g x)) x change ε ≤ dist (f' i x) (g' x) ↔ ε ≤ dist (f i x) (g x) case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i x : α hxf : f i x = f' i x hxg : g x = g' x ⊢ ε ≤ dist (f' i x) (g' x) ↔ ε ≤ dist (f i x) (g x) rw [hxg, hxf] α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε this : (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) =ᶠ[l] fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)} ⊢ Tendsto (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) l (𝓝 0) rw [tendsto_congr' this] α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε this : (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) =ᶠ[l] fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)} ⊢ Tendsto (fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)}) l (𝓝 0) exact h_tendsto ε hε ** Qed
MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ⊢ TendstoInMeasure μ f atTop g refine' fun ε hε => ENNReal.tendsto_atTop_zero.mpr fun δ hδ => _ α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0∞ hδ : δ > 0 ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ δ by_cases hδi : δ = ∞ case neg α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0∞ hδ : δ > 0 hδi : ¬δ = ⊤ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ δ lift δ to ℝ≥0 using hδi case neg.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : ↑δ > 0 ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ rw [gt_iff_lt, ENNReal.coe_pos, ← NNReal.coe_pos] at hδ case neg.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ obtain ⟨t, _, ht, hunif⟩ := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg hδ case neg.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ENNReal.ofReal ↑δ hunif : TendstoUniformlyOn (fun n => f n) g atTop tᶜ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ rw [ENNReal.ofReal_coe_nnreal] at ht case neg.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : TendstoUniformlyOn (fun n => f n) g atTop tᶜ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ rw [Metric.tendstoUniformlyOn_iff] at hunif case neg.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ obtain ⟨N, hN⟩ := eventually_atTop.1 (hunif ε hε) case neg.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ refine' ⟨N, fun n hn => _⟩ case neg.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ suffices : { x : α \| ε ≤ dist (f n x) (g x) } ⊆ t case neg.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N this : {x \| ε ≤ dist (f n x) (g x)} ⊆ t ⊢ ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ {x \| ε ≤ dist (f n x) (g x)} ⊆ t exact (measure_mono this).trans ht case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ {x \| ε ≤ dist (f n x) (g x)} ⊆ t rw [← Set.compl_subset_compl] case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ tᶜ ⊆ {x \| ε ≤ dist (f n x) (g x)}ᶜ intro x hx case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N x : α hx : x ∈ tᶜ ⊢ x ∈ {x \| ε ≤ dist (f n x) (g x)}ᶜ rw [Set.mem_compl_iff, Set.nmem_setOf_iff, dist_comm, not_le] case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N x : α hx : x ∈ tᶜ ⊢ dist (g x) (f n x) < ε exact hN n hn x hx case pos α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0∞ hδ : δ > 0 hδi : δ = ⊤ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ δ simp only [hδi, imp_true_iff, le_top, exists_const] ** Qed
MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g n : ℕ ⊢ ∃ N, ∀ (m_1 : ℕ), m_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n specialize hfg ((2⁻¹ : ℝ) ^ n) (by simp only [Real.rpow_nat_cast, inv_pos, zero_lt_two, pow_pos]) α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E n : ℕ hfg : Tendsto (fun i => ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f i x) (g x)}) atTop (𝓝 0) ⊢ ∃ N, ∀ (m_1 : ℕ), m_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n rw [ENNReal.tendsto_atTop_zero] at hfg α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E n : ℕ hfg : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n_1 : ℕ), n_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f n_1 x) (g x)} ≤ ε ⊢ ∃ N, ∀ (m_1 : ℕ), m_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n exact hfg ((2 : ℝ≥0∞)⁻¹ ^ n) (pos_iff_ne_zero.mpr fun h_zero => by simpa using pow_eq_zero h_zero) α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g n : ℕ ⊢ 0 < 2⁻¹ ^ n simp only [Real.rpow_nat_cast, inv_pos, zero_lt_two, pow_pos] α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E n : ℕ hfg : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n_1 : ℕ), n_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f n_1 x) (g x)} ≤ ε h_zero : 2⁻¹ ^ n = 0 ⊢ False simpa using pow_eq_zero h_zero ** Qed
MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g n k : ℕ hn : seqTendstoAeSeq hfg n ≤ k ⊢ ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ n cases n case zero α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g k : ℕ hn : seqTendstoAeSeq hfg Nat.zero ≤ k ⊢ ↑↑μ {x \| 2⁻¹ ^ Nat.zero ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ Nat.zero exact Classical.choose_spec (exists_nat_measure_lt_two_inv hfg 0) k hn case succ α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g k n✝ : ℕ hn : seqTendstoAeSeq hfg (Nat.succ n✝) ≤ k ⊢ ↑↑μ {x \| 2⁻¹ ^ Nat.succ n✝ ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ Nat.succ n✝ exact Classical.choose_spec (exists_nat_measure_lt_two_inv hfg _) _ (le_trans (le_max_left _ _) hn) ** Qed
MeasureTheory.TendstoInMeasure.exists_seq_tendstoInMeasure_atTop α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ⊢ ∃ ns, TendstoInMeasure μ (fun n => f (ns n)) atTop g obtain ⟨ns, h_tendsto_ns⟩ : ∃ ns : ℕ → ι, Tendsto ns atTop u := exists_seq_tendsto u case intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ns : ℕ → ι h_tendsto_ns : Tendsto ns atTop u ⊢ ∃ ns, TendstoInMeasure μ (fun n => f (ns n)) atTop g exact ⟨ns, fun ε hε => (hfg ε hε).comp h_tendsto_ns⟩ ** Qed
MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae' α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ⊢ ∃ ns, ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) obtain ⟨ms, hms⟩ := hfg.exists_seq_tendstoInMeasure_atTop case intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ms : ℕ → ι hms : TendstoInMeasure μ (fun n => f (ms n)) atTop g ⊢ ∃ ns, ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) obtain ⟨ns, -, hns⟩ := hms.exists_seq_tendsto_ae case intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ms : ℕ → ι hms : TendstoInMeasure μ (fun n => f (ms n)) atTop g ns : ℕ → ℕ hns : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ms (ns i)) x) atTop (𝓝 (g x)) ⊢ ∃ ns, ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) exact ⟨ms ∘ ns, hns⟩ ** Qed
MeasureTheory.TendstoInMeasure.aemeasurable α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : BorelSpace E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : TendstoInMeasure μ f u g ⊢ AEMeasurable g obtain ⟨ns, hns⟩ := h_tendsto.exists_seq_tendsto_ae' case intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : BorelSpace E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : TendstoInMeasure μ f u g ns : ℕ → ι hns : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) ⊢ AEMeasurable g exact aemeasurable_of_tendsto_metrizable_ae atTop (fun n => hf (ns n)) hns ** Qed
MeasureTheory.AEEqFun.mk_coeFn α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = f conv_rhs => rw [← Quotient.out_eq' f] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' (Quotient.out' f) set g : { f : α → β // AEStronglyMeasurable f μ } := Quotient.out' f α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : { f // AEStronglyMeasurable f μ } := Quotient.out' f ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g have : g = ⟨g.1, g.2⟩ := Subtype.eq rfl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : { f // AEStronglyMeasurable f μ } := Quotient.out' f this : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) } ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g rw [this, ← mk, mk_eq_mk] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : { f // AEStronglyMeasurable f μ } := Quotient.out' f this : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) } ⊢ ↑f =ᵐ[μ] ↑g exact (AEStronglyMeasurable.ae_eq_mk _).symm ** Qed
MeasureTheory.AEEqFun.ext α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : α →ₘ[μ] β h : ↑f =ᵐ[μ] ↑g ⊢ f = g rwa [← f.mk_coeFn, ← g.mk_coeFn, mk_eq_mk] ** Qed
MeasureTheory.AEEqFun.ext_iff α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : α →ₘ[μ] β h : f = g ⊢ ↑f =ᵐ[μ] ↑g rw [h] ** Qed
MeasureTheory.AEEqFun.coeFn_mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α → β hf : AEStronglyMeasurable f μ ⊢ ↑(mk f hf) =ᵐ[μ] f apply (AEStronglyMeasurable.ae_eq_mk _).symm.trans α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α → β hf : AEStronglyMeasurable f μ ⊢ ↑(Quotient.out' (mk f hf)) =ᵐ[μ] f exact @Quotient.mk_out' _ (μ.aeEqSetoid β) (⟨f, hf⟩ : { f // AEStronglyMeasurable f μ }) ** Qed
MeasureTheory.AEEqFun.comp_eq_mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β → γ hg : Continuous g f : α →ₘ[μ] β ⊢ comp g hg f = mk (g ∘ ↑f) (_ : AEStronglyMeasurable (fun x => g (↑f x)) μ) rw [← comp_mk g hg f f.aestronglyMeasurable, mk_coeFn] ** Qed
MeasureTheory.AEEqFun.compMeasurable_eq_mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹⁰ : MeasurableSpace α μ ν : Measure α inst✝⁹ : TopologicalSpace β inst✝⁸ : TopologicalSpace γ inst✝⁷ : TopologicalSpace δ inst✝⁶ : MeasurableSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : BorelSpace β inst✝³ : MeasurableSpace γ inst✝² : PseudoMetrizableSpace γ inst✝¹ : OpensMeasurableSpace γ inst✝ : SecondCountableTopology γ g : β → γ hg : Measurable g f : α →ₘ[μ] β ⊢ compMeasurable g hg f = mk (g ∘ ↑f) (_ : AEStronglyMeasurable (g ∘ ↑f) μ) rw [← compMeasurable_mk g hg f f.aestronglyMeasurable, mk_coeFn] ** Qed
MeasureTheory.AEEqFun.coeFn_pair α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : α →ₘ[μ] γ ⊢ ↑(pair f g) =ᵐ[μ] fun x => (↑f x, ↑g x) rw [pair_eq_mk] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : α →ₘ[μ] γ ⊢ ↑(mk (fun x => (↑f x, ↑g x)) (_ : AEStronglyMeasurable (fun x => (↑f x, ↑g x)) μ)) =ᵐ[μ] fun x => (↑f x, ↑g x) apply coeFn_mk ** Qed
MeasureTheory.AEEqFun.toGerm_injective α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : α →ₘ[μ] β H : toGerm f = toGerm g ⊢ ↑↑f = ↑↑g rwa [← toGerm_eq, ← toGerm_eq] ** Qed
MeasureTheory.AEEqFun.comp_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β → γ hg : Continuous g f✝ : α →ₘ[μ] β f : α → β x✝ : AEStronglyMeasurable f μ ⊢ toGerm (comp g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝)) simp ** Qed
MeasureTheory.AEEqFun.compMeasurable_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹⁰ : MeasurableSpace α μ ν : Measure α inst✝⁹ : TopologicalSpace β inst✝⁸ : TopologicalSpace γ inst✝⁷ : TopologicalSpace δ inst✝⁶ : MeasurableSpace β inst✝⁵ : BorelSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : PseudoMetrizableSpace γ inst✝² : SecondCountableTopology γ inst✝¹ : MeasurableSpace γ inst✝ : OpensMeasurableSpace γ g : β → γ hg : Measurable g f✝ : α →ₘ[μ] β f : α → β x✝ : AEStronglyMeasurable f μ ⊢ toGerm (compMeasurable g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝)) simp ** Qed
MeasureTheory.AEEqFun.comp₂_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β → γ → δ hg : Continuous (uncurry g) f₁✝ : α →ₘ[μ] β f₂✝ : α →ₘ[μ] γ f₁ : α → β x✝¹ : AEStronglyMeasurable f₁ μ f₂ : α → γ x✝ : AEStronglyMeasurable f₂ μ ⊢ toGerm (comp₂ g hg (mk f₁ x✝¹) (mk f₂ x✝)) = Germ.map₂ g (toGerm (mk f₁ x✝¹)) (toGerm (mk f₂ x✝)) simp ** Qed
MeasureTheory.AEEqFun.comp₂Measurable_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹⁴ : MeasurableSpace α μ ν : Measure α inst✝¹³ : TopologicalSpace β inst✝¹² : TopologicalSpace γ inst✝¹¹ : TopologicalSpace δ inst✝¹⁰ : PseudoMetrizableSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : PseudoMetrizableSpace γ inst✝⁶ : SecondCountableTopologyEither β γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : PseudoMetrizableSpace δ inst✝² : SecondCountableTopology δ inst✝¹ : MeasurableSpace δ inst✝ : OpensMeasurableSpace δ g : β → γ → δ hg : Measurable (uncurry g) f₁✝ : α →ₘ[μ] β f₂✝ : α →ₘ[μ] γ f₁ : α → β x✝¹ : AEStronglyMeasurable f₁ μ f₂ : α → γ x✝ : AEStronglyMeasurable f₂ μ ⊢ toGerm (comp₂Measurable g hg (mk f₁ x✝¹) (mk f₂ x✝)) = Germ.map₂ g (toGerm (mk f₁ x✝¹)) (toGerm (mk f₂ x✝)) simp ** Qed
MeasureTheory.AEEqFun.le_sup_left α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ f ≤ f ⊔ g rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ ↑f ≤ᵐ[μ] ↑(f ⊔ g) filter_upwards [coeFn_sup f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑f a✝ ≤ ↑(f ⊔ g) a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑f a✝ ≤ ↑f a✝ ⊔ ↑g a✝ exact le_sup_left ** Qed
MeasureTheory.AEEqFun.le_sup_right α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ g ≤ f ⊔ g rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ ↑g ≤ᵐ[μ] ↑(f ⊔ g) filter_upwards [coeFn_sup f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑g a✝ ≤ ↑(f ⊔ g) a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑g a✝ ≤ ↑f a✝ ⊔ ↑g a✝ exact le_sup_right ** Qed
MeasureTheory.AEEqFun.sup_le α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : f ≤ f' hg : g ≤ f' ⊢ f ⊔ g ≤ f' rw [← coeFn_le] at hf hg ⊢ α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : ↑f ≤ᵐ[μ] ↑f' hg : ↑g ≤ᵐ[μ] ↑f' ⊢ ↑(f ⊔ g) ≤ᵐ[μ] ↑f' filter_upwards [hf, hg, coeFn_sup f g] with _ haf hag ha_sup case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : ↑f ≤ᵐ[μ] ↑f' hg : ↑g ≤ᵐ[μ] ↑f' a✝ : α haf : ↑f a✝ ≤ ↑f' a✝ hag : ↑g a✝ ≤ ↑f' a✝ ha_sup : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑(f ⊔ g) a✝ ≤ ↑f' a✝ rw [ha_sup] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : ↑f ≤ᵐ[μ] ↑f' hg : ↑g ≤ᵐ[μ] ↑f' a✝ : α haf : ↑f a✝ ≤ ↑f' a✝ hag : ↑g a✝ ≤ ↑f' a✝ ha_sup : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑f a✝ ⊔ ↑g a✝ ≤ ↑f' a✝ exact sup_le haf hag ** Qed
MeasureTheory.AEEqFun.inf_le_left α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ f ⊓ g ≤ f rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑f filter_upwards [coeFn_inf f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝ exact inf_le_left ** Qed
MeasureTheory.AEEqFun.inf_le_right α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ f ⊓ g ≤ g rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑g filter_upwards [coeFn_inf f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑g a✝ exact inf_le_right ** Qed
MeasureTheory.AEEqFun.le_inf α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : f' ≤ f hg : f' ≤ g ⊢ f' ≤ f ⊓ g rw [← coeFn_le] at hf hg ⊢ α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : ↑f' ≤ᵐ[μ] ↑f hg : ↑f' ≤ᵐ[μ] ↑g ⊢ ↑f' ≤ᵐ[μ] ↑(f ⊓ g) filter_upwards [hf, hg, coeFn_inf f g] with _ haf hag ha_inf case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : ↑f' ≤ᵐ[μ] ↑f hg : ↑f' ≤ᵐ[μ] ↑g a✝ : α haf : ↑f' a✝ ≤ ↑f a✝ hag : ↑f' a✝ ≤ ↑g a✝ ha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f' a✝ ≤ ↑(f ⊓ g) a✝ rw [ha_inf] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : ↑f' ≤ᵐ[μ] ↑f hg : ↑f' ≤ᵐ[μ] ↑g a✝ : α haf : ↑f' a✝ ≤ ↑f a✝ hag : ↑f' a✝ ≤ ↑g a✝ ha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f' a✝ ≤ ↑f a✝ ⊓ ↑g a✝ exact le_inf haf hag ** Qed
MeasureTheory.AEEqFun.lintegral_add α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f✝ g✝ : α →ₘ[μ] ℝ≥0∞ f : α → ℝ≥0∞ hf : AEStronglyMeasurable f μ g : α → ℝ≥0∞ x✝ : AEStronglyMeasurable g μ ⊢ lintegral (mk f hf + mk g x✝) = lintegral (mk f hf) + lintegral (mk g x✝) simp [lintegral_add_left' hf.aemeasurable] ** Qed
MeasureTheory.Measure.haveLebesgueDecomposition_spec α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ Measurable (rnDeriv μ ν) ∧ singularPart μ ν ⟂ₘ ν ∧ μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) rw [singularPart, rnDeriv, dif_pos h, dif_pos h] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ Measurable (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).2 ∧ (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).1 ⟂ₘ ν ∧ μ = (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).1 + withDensity ν (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).2 exact Classical.choose_spec h.lebesgue_decomposition ** Qed
MeasureTheory.Measure.measurable_rnDeriv α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ Measurable (rnDeriv μ ν) by_cases h : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ Measurable (rnDeriv μ ν) exact (haveLebesgueDecomposition_spec μ ν).1 case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ Measurable (rnDeriv μ ν) rw [rnDeriv, dif_neg h] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ Measurable 0 exact measurable_zero ** Qed
MeasureTheory.Measure.mutuallySingular_singularPart α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ singularPart μ ν ⟂ₘ ν by_cases h : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ⟂ₘ ν exact (haveLebesgueDecomposition_spec μ ν).2.1 case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ⟂ₘ ν rw [singularPart, dif_neg h] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ 0 ⟂ₘ ν exact MutuallySingular.zero_left ** Qed
MeasureTheory.Measure.singularPart_le α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ singularPart μ ν ≤ μ by_cases hl : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ≤ μ cases' (haveLebesgueDecomposition_spec μ ν).2 with _ h case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ singularPart μ ν ≤ μ conv_rhs => rw [h] case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ singularPart μ ν ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν) exact Measure.le_add_right le_rfl case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ≤ μ rw [singularPart, dif_neg hl] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ 0 ≤ μ exact Measure.zero_le μ ** Qed
MeasureTheory.Measure.withDensity_rnDeriv_le α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ withDensity ν (rnDeriv μ ν) ≤ μ by_cases hl : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν ⊢ withDensity ν (rnDeriv μ ν) ≤ μ cases' (haveLebesgueDecomposition_spec μ ν).2 with _ h case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ withDensity ν (rnDeriv μ ν) ≤ μ conv_rhs => rw [h] case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ withDensity ν (rnDeriv μ ν) ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν) exact Measure.le_add_left le_rfl case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ withDensity ν (rnDeriv μ ν) ≤ μ rw [rnDeriv, dif_neg hl, withDensity_zero] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ 0 ≤ μ exact Measure.zero_le μ ** Qed
MeasureTheory.Measure.lintegral_rnDeriv_lt_top_of_measure_ne_top α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ by_cases hl : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl : HaveLebesgueDecomposition μ ν ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ haveI := hl case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ obtain ⟨-, -, hadd⟩ := haveLebesgueDecomposition_spec μ ν case pos.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ suffices : (∫⁻ x in toMeasurable μ s, μ.rnDeriv ν x ∂ν) < ∞ case pos.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this✝ : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) this : ∫⁻ (x : α) in toMeasurable μ s, rnDeriv μ ν x ∂ν < ⊤ ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ∫⁻ (x : α) in toMeasurable μ s, rnDeriv μ ν x ∂ν < ⊤ exact lt_of_le_of_lt (lintegral_mono_set (subset_toMeasurable _ _)) this case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ∫⁻ (x : α) in toMeasurable μ s, rnDeriv μ ν x ∂ν < ⊤ rw [← withDensity_apply _ (measurableSet_toMeasurable _ _)] case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ↑↑(withDensity ν fun x => rnDeriv μ ν x) (toMeasurable μ s) < ⊤ refine' lt_of_le_of_lt (le_add_left le_rfl : _ ≤ μ.singularPart ν (toMeasurable μ s) + ν.withDensity (μ.rnDeriv ν) (toMeasurable μ s)) _ case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ↑↑(singularPart μ ν) (toMeasurable μ s) + ↑↑(withDensity ν (rnDeriv μ ν)) (toMeasurable μ s) < ⊤ rw [← Measure.add_apply, ← hadd, measure_toMeasurable] case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ↑↑μ s < ⊤ exact hs.lt_top case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl : ¬HaveLebesgueDecomposition μ ν ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ erw [Measure.rnDeriv, dif_neg hl, lintegral_zero] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl : ¬HaveLebesgueDecomposition μ ν ⊢ 0 < ⊤ exact WithTop.zero_lt_top ** Qed
MeasureTheory.Measure.lintegral_rnDeriv_lt_top α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : IsFiniteMeasure μ ⊢ ∫⁻ (x : α), rnDeriv μ ν x ∂ν < ⊤ rw [← set_lintegral_univ] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : IsFiniteMeasure μ ⊢ ∫⁻ (x : α) in univ, rnDeriv μ ν x ∂ν < ⊤ exact lintegral_rnDeriv_lt_top_of_measure_ne_top _ (measure_lt_top _ _).ne ** Qed
MeasureTheory.Measure.rnDeriv_lt_top α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ ⊢ ∀ᵐ (x : α) ∂ν, rnDeriv μ ν x < ⊤ suffices ∀ n, ∀ᵐ x ∂ν, x ∈ spanningSets μ n → μ.rnDeriv ν x < ∞ by filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ ⊢ ∀ (n : ℕ), ∀ᵐ (x : α) ∂ν, x ∈ spanningSets μ n → rnDeriv μ ν x < ⊤ intro n α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ∀ᵐ (x : α) ∂ν, x ∈ spanningSets μ n → rnDeriv μ ν x < ⊤ rw [← ae_restrict_iff' (measurable_spanningSets _ _)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ∀ᵐ (x : α) ∂restrict ν (spanningSets μ n), rnDeriv μ ν x < ⊤ apply ae_lt_top (measurable_rnDeriv _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ∫⁻ (x : α) in spanningSets μ n, rnDeriv μ ν x ∂ν ≠ ⊤ refine' (lintegral_rnDeriv_lt_top_of_measure_ne_top _ _).ne α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ↑↑μ (spanningSets μ n) ≠ ⊤ exact (measure_spanningSets_lt_top _ _).ne α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ this : ∀ (n : ℕ), ∀ᵐ (x : α) ∂ν, x ∈ spanningSets μ n → rnDeriv μ ν x < ⊤ ⊢ ∀ᵐ (x : α) ∂ν, rnDeriv μ ν x < ⊤ filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _) ** Qed
MeasureTheory.Measure.eq_singularPart α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f ⊢ s = singularPart μ ν haveI : HaveLebesgueDecomposition μ ν := ⟨⟨⟨s, f⟩, hf, hs, hadd⟩⟩ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν ⊢ s = singularPart μ ν obtain ⟨hmeas, hsing, hadd'⟩ := haveLebesgueDecomposition_spec μ ν case intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hsing : singularPart μ ν ⟂ₘ ν hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ s = singularPart μ ν obtain ⟨⟨S, hS₁, hS₂, hS₃⟩, ⟨T, hT₁, hT₂, hT₃⟩⟩ := hs, hsing case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ s = singularPart μ ν rw [hadd'] at hadd case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ s = singularPart μ ν have hνinter : ν (S ∩ T)ᶜ = 0 := by rw [compl_inter] refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) rw [hT₃, hS₃, add_zero] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ s = singularPart μ ν have heq : s.restrict (S ∩ T)ᶜ = (μ.singularPart ν).restrict (S ∩ T)ᶜ := by ext1 A hA have hf : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) have hrn : ν.withDensity (μ.rnDeriv ν) (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) rw [restrict_apply hA, restrict_apply hA, ← add_zero (s (A ∩ (S ∩ T)ᶜ)), ← hf, ← add_apply, ← hadd, add_apply, hrn, add_zero] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ ⊢ s = singularPart μ ν have heq' : ∀ A : Set α, MeasurableSet A → s A = s.restrict (S ∩ T)ᶜ A := by intro A hA have hsinter : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) rw [restrict_apply hA, ← diff_eq, AEDisjoint.measure_diff_left hsinter] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A ⊢ s = singularPart μ ν ext1 A hA case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A ⊢ ↑↑s A = ↑↑(singularPart μ ν) A have hμinter : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A hμinter : ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 ⊢ ↑↑s A = ↑↑(singularPart μ ν) A rw [heq' A hA, heq, restrict_apply hA, ← diff_eq, AEDisjoint.measure_diff_left hμinter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (S ∩ T)ᶜ = 0 rw [compl_inter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (Sᶜ ∪ Tᶜ) = 0 refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν Sᶜ + ↑↑ν Tᶜ ≤ 0 rw [hT₃, hS₃, add_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ ext1 A hA case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑(restrict s (S ∩ T)ᶜ) A = ↑↑(restrict (singularPart μ ν) (S ∩ T)ᶜ) A have hf : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(restrict s (S ∩ T)ᶜ) A = ↑↑(restrict (singularPart μ ν) (S ∩ T)ᶜ) A have hrn : ν.withDensity (μ.rnDeriv ν) (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 hrn : ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(restrict s (S ∩ T)ᶜ) A = ↑↑(restrict (singularPart μ ν) (S ∩ T)ᶜ) A rw [restrict_apply hA, restrict_apply hA, ← add_zero (s (A ∩ (S ∩ T)ᶜ)), ← hf, ← add_apply, ← hadd, add_apply, hrn, add_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 refine' withDensity_absolutelyContinuous ν _ _ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) ≤ 0 exact hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 refine' withDensity_absolutelyContinuous ν _ _ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) ≤ 0 exact hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ ⊢ ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A intro A hA α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A ⊢ ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A have hsinter : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A hsinter : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A rw [restrict_apply hA, ← diff_eq, AEDisjoint.measure_diff_left hsinter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) ≤ 0 exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) ≤ 0 exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) ** Qed
MeasureTheory.Measure.singularPart_zero α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ ν : Measure α ⊢ singularPart 0 ν = 0 refine' (eq_singularPart measurable_zero MutuallySingular.zero_left _).symm α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ ν : Measure α ⊢ 0 = 0 + withDensity ν 0 rw [zero_add, withDensity_zero] ** Qed
MeasureTheory.Measure.singularPart_add α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ singularPart (μ₁ + μ₂) ν = singularPart μ₁ ν + singularPart μ₂ ν refine' (eq_singularPart ((measurable_rnDeriv μ₁ ν).add (measurable_rnDeriv μ₂ ν)) ((haveLebesgueDecomposition_spec _ _).2.1.add_left (haveLebesgueDecomposition_spec _ _).2.1) _).symm α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ μ₁ + μ₂ = singularPart μ₁ ν + singularPart μ₂ ν + withDensity ν fun a => rnDeriv μ₁ ν a + rnDeriv μ₂ ν a erw [withDensity_add_left (measurable_rnDeriv μ₁ ν)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ μ₁ + μ₂ = singularPart μ₁ ν + singularPart μ₂ ν + (withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a) conv_rhs => rw [add_assoc, add_comm (μ₂.singularPart ν), ← add_assoc, ← add_assoc] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ μ₁ + μ₂ = (singularPart μ₁ ν + withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a) + singularPart μ₂ ν rw [← haveLebesgueDecomposition_add μ₁ ν, add_assoc, add_comm (ν.withDensity (μ₂.rnDeriv ν)), ← haveLebesgueDecomposition_add μ₂ ν] ** Qed
MeasureTheory.Measure.singularPart_withDensity α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f ⊢ withDensity ν f = 0 + withDensity ν f rw [zero_add] ** Qed
MeasureTheory.Measure.eq_withDensity_rnDeriv α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) haveI : HaveLebesgueDecomposition μ ν := ⟨⟨⟨s, f⟩, hf, hs, hadd⟩⟩ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) obtain ⟨hmeas, hsing, hadd'⟩ := haveLebesgueDecomposition_spec μ ν case intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hsing : singularPart μ ν ⟂ₘ ν hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) obtain ⟨⟨S, hS₁, hS₂, hS₃⟩, ⟨T, hT₁, hT₂, hT₃⟩⟩ := hs, hsing case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) rw [hadd'] at hadd case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) have hνinter : ν (S ∩ T)ᶜ = 0 := by rw [compl_inter] refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) rw [hT₃, hS₃, add_zero] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) have heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) := by ext1 A hA have hs : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) have hsing : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ← add_apply, add_comm, ← hadd, add_apply, hsing, zero_add] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) have heq' : ∀ A : Set α, MeasurableSet A → ν.withDensity f A = (ν.withDensity f).restrict (S ∩ T) A := by intro A hA have hνfinter : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by rw [← nonpos_iff_eq_zero] exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono (inter_subset_right _ _) rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq, measure_inter_add_diff _ (hS₁.inter hT₁)] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) ext1 A hA case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) A = ↑↑(withDensity ν (rnDeriv μ ν)) A have hνrn : ν.withDensity (μ.rnDeriv ν) (A ∩ (S ∩ T)ᶜ) = 0 := by rw [← nonpos_iff_eq_zero] exact withDensity_absolutelyContinuous ν (μ.rnDeriv ν) hνinter ▸ measure_mono (inter_subset_right _ _) case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A hνrn : ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(withDensity ν f) A = ↑↑(withDensity ν (rnDeriv μ ν)) A rw [heq' A hA, heq, ← add_zero ((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) A), ← hνrn, restrict_apply hA, ← diff_eq, measure_inter_add_diff _ (hS₁.inter hT₁)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (S ∩ T)ᶜ = 0 rw [compl_inter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (Sᶜ ∪ Tᶜ) = 0 refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν Sᶜ + ↑↑ν Tᶜ ≤ 0 rw [hT₃, hS₃, add_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) ext1 A hA case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑(restrict (withDensity ν f) (S ∩ T)) A = ↑↑(restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T)) A have hs : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(restrict (withDensity ν f) (S ∩ T)) A = ↑↑(restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T)) A have hsing : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 hsing : ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(restrict (withDensity ν f) (S ∩ T)) A = ↑↑(restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T)) A rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ← add_apply, add_comm, ← hadd, add_apply, hsing, zero_add] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) ≤ 0 exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) ≤ 0 exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) ⊢ ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A intro A hA α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A have hνfinter : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by rw [← nonpos_iff_eq_zero] exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A hνfinter : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq, measure_inter_add_diff _ (hS₁.inter hT₁)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) ≤ 0 exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) ≤ 0 exact withDensity_absolutelyContinuous ν (μ.rnDeriv ν) hνinter ▸ measure_mono (inter_subset_right _ _) ** Qed
MeasureTheory.Measure.LebesgueDecomposition.zero_mem_measurableLE α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α A : Set α x✝ : MeasurableSet A ⊢ ∫⁻ (x : α) in A, OfNat.ofNat 0 x ∂μ ≤ ↑↑ν A simp ** Qed
MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a refine Option.ext fun x => ?_ α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 ⊢ x ∈ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ↔ x ∈ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a simp only [Option.mem_def, ENNReal.some_eq_coe] α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = ↑x ↔ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x constructor <;> intro h <;> rw [← h] case mp α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = ↑x ⊢ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a symm case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a set c := ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a with hc case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ⊢ c = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a set d := f m.succ a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a with hd case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ c = d rw [@le_antisymm_iff ℝ≥0∞, hc, hd] case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ∧ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a refine' ⟨_, _⟩ case mpr.refine'_1 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a refine' iSup₂_le fun n hn => _ case mpr.refine'_1 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a n : ℕ hn : n ≤ m + 1 ⊢ f n a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a rcases Nat.of_le_succ hn with (h \| h) case mpr.refine'_1.inl α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h✝ : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a n : ℕ hn : n ≤ m + 1 h : n ≤ m ⊢ f n a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a exact le_sup_of_le_right (le_iSup₂ (f := fun k (_ : k ≤ m) => f k a) n h) case mpr.refine'_1.inr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h✝ : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a n : ℕ hn : n ≤ m + 1 h : n = Nat.succ m ⊢ f n a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a exact h ▸ le_sup_left case mpr.refine'_2 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a refine' sup_le _ (biSup_mono fun n hn => hn.trans m.le_succ) case mpr.refine'_2 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ f (Nat.succ m) a ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a convert @le_iSup₂ ℝ≥0∞ ℕ (fun i => i ≤ m + 1) _ _ m.succ le_rfl case h.e'_3 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ f (Nat.succ m) a = f (Nat.succ m) a rfl ** Qed
MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE' α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν n : ℕ ⊢ ⨆ k, ⨆ (_ : k ≤ n), f k ∈ measurableLE μ ν convert iSup_mem_measurableLE f hf n case h.e'_4.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν n : ℕ x✝ : α ⊢ iSup (fun k => ⨆ (_ : k ≤ n), f k) x✝ = ⨆ k, ⨆ (_ : k ≤ n), f k x✝ refine Option.ext fun x => ?_ case h.e'_4.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν n : ℕ x✝ : α x : ℝ≥0 ⊢ x ∈ iSup (fun k => ⨆ (_ : k ≤ n), f k) x✝ ↔ x ∈ ⨆ k, ⨆ (_ : k ≤ n), f k x✝ simp ** Qed
MeasureTheory.SignedMeasure.measurable_rnDeriv α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Measurable (rnDeriv s μ) rw [rnDeriv] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart μ x) - ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) measurability ** Qed
MeasureTheory.SignedMeasure.integrable_rnDeriv case refine'_2 α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) constructor case refine'_2.right α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ HasFiniteIntegral fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) exact hasFiniteIntegral_toReal_of_lintegral_ne_top (lintegral_rnDeriv_lt_top _ μ).ne case refine'_2.left α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x)) μ apply Measurable.aestronglyMeasurable case refine'_2.left.hf α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) measurability ** Qed

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.

formal stringlengths 41 427k	informal stringclasses 1 value
MonotoneOn.integrableOn_isCompact X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹¹ : MeasurableSpace X inst✝¹⁰ : TopologicalSpace X inst✝⁹ : MeasurableSpace Y inst✝⁸ : TopologicalSpace Y inst✝⁷ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁶ : BorelSpace X inst✝⁵ : ConditionallyCompleteLinearOrder X inst✝⁴ : ConditionallyCompleteLinearOrder E inst✝³ : OrderTopology X inst✝² : OrderTopology E inst✝¹ : SecondCountableTopology E inst✝ : IsFiniteMeasureOnCompacts μ hs : IsCompact s hmono : MonotoneOn f s ⊢ IntegrableOn f s obtain rfl \| h := s.eq_empty_or_nonempty case inl X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹¹ : MeasurableSpace X inst✝¹⁰ : TopologicalSpace X inst✝⁹ : MeasurableSpace Y inst✝⁸ : TopologicalSpace Y inst✝⁷ : NormedAddCommGroup E f g : X → E μ : Measure X inst✝⁶ : BorelSpace X inst✝⁵ : ConditionallyCompleteLinearOrder X inst✝⁴ : ConditionallyCompleteLinearOrder E inst✝³ : OrderTopology X inst✝² : OrderTopology E inst✝¹ : SecondCountableTopology E inst✝ : IsFiniteMeasureOnCompacts μ hs : IsCompact ∅ hmono : MonotoneOn f ∅ ⊢ IntegrableOn f ∅ exact integrableOn_empty case inr X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝¹¹ : MeasurableSpace X inst✝¹⁰ : TopologicalSpace X inst✝⁹ : MeasurableSpace Y inst✝⁸ : TopologicalSpace Y inst✝⁷ : NormedAddCommGroup E f g : X → E μ : Measure X s : Set X inst✝⁶ : BorelSpace X inst✝⁵ : ConditionallyCompleteLinearOrder X inst✝⁴ : ConditionallyCompleteLinearOrder E inst✝³ : OrderTopology X inst✝² : OrderTopology E inst✝¹ : SecondCountableTopology E inst✝ : IsFiniteMeasureOnCompacts μ hs : IsCompact s hmono : MonotoneOn f s h : Set.Nonempty s ⊢ IntegrableOn f s exact hmono.integrableOn_of_measure_ne_top (hs.isLeast_sInf h) (hs.isGreatest_sSup h) hs.measure_lt_top.ne hs.measurableSet ** Qed
MeasureTheory.IntegrableOn.mul_continuousOn_of_subset ** 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : IntegrableOn g A hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K ⊢ IntegrableOn (fun x => g x * g' x) A rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : IntegrableOn g A hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C ⊢ IntegrableOn (fun x => g x * g' x) A rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C ⊢ Memℒp (fun x => g x * g' x) 1 ** have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by filter_upwards [ae_restrict_mem hA] with x hx refine' (norm_mul_le _ _).trans _ rw [mul_comm] apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) ** case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖ ⊢ Memℒp (fun x => g x * g' x) 1 exact Memℒp.of_le_mul hg (hg.aestronglyMeasurable.mul <\| (hg'.mono hAK).aestronglyMeasurable hA) this 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C ⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖ filter_upwards [ae_restrict_mem hA] with x hx case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x * g' x‖ ≤ C * ‖g x‖ refine' (norm_mul_le _ _).trans _ case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g x‖ rw [mul_comm] case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : Memℒp g 1 hg' : ContinuousOn g' K hA : MeasurableSet A hK : IsCompact K hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g' x‖ * ‖g x‖ ≤ C * ‖g x‖ apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) Qed
MeasureTheory.IntegrableOn.continuousOn_mul_of_subset ** 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : IntegrableOn g' A hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K ⊢ IntegrableOn (fun x => g x * g' x) A rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : IntegrableOn g' A hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C ⊢ IntegrableOn (fun x => g x * g' x) A rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢ case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C ⊢ Memℒp (fun x => g x * g' x) 1 ** have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by filter_upwards [ae_restrict_mem hA] with x hx refine' (norm_mul_le _ _).trans _ apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) ** case intro 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖ ⊢ Memℒp (fun x => g x * g' x) 1 exact Memℒp.of_le_mul hg' (((hg.mono hAK).aestronglyMeasurable hA).mul hg'.aestronglyMeasurable) this 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C ⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖ filter_upwards [ae_restrict_mem hA] with x hx case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖ refine' (norm_mul_le _ _).trans _ case h 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 A K : Set X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R g g' : X → R hg : ContinuousOn g K hg' : Memℒp g' 1 hK : IsCompact K hA : MeasurableSet A hAK : A ⊆ K C : ℝ hC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C x : X hx : x ∈ A ⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖ apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _) Qed
MeasureTheory.IntegrableOn.continuousOn_smul X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ IntegrableOn (fun x => f x • g x) K rw [IntegrableOn, ← integrable_norm_iff] X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ Integrable fun a => ‖f a • g a‖ simp_rw [norm_smul] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ Integrable fun a => ‖f a‖ * ‖g a‖ refine' IntegrableOn.continuousOn_mul _ hg.norm hK X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ ContinuousOn (fun a => ‖f a‖) K exact continuous_norm.comp_continuousOn hf X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X 𝕜 g : X → E hg : IntegrableOn g K f : X → 𝕜 hf : ContinuousOn f K hK : IsCompact K ⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K) exact (hf.aestronglyMeasurable hK.measurableSet).smul hg.1 Qed
MeasureTheory.IntegrableOn.smul_continuousOn X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ IntegrableOn (fun x => f x • g x) K rw [IntegrableOn, ← integrable_norm_iff] X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ Integrable fun a => ‖f a • g a‖ simp_rw [norm_smul] ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ Integrable fun a => ‖f a‖ * ‖g a‖ refine' IntegrableOn.mul_continuousOn hf.norm _ hK X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ ContinuousOn (fun a => ‖g a‖) K exact continuous_norm.comp_continuousOn hg X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X 𝕜 : Type u_5 inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : T2Space X inst✝ : SecondCountableTopologyEither X E f : X → 𝕜 hf : IntegrableOn f K g : X → E hg : ContinuousOn g K hK : IsCompact K ⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K) exact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet) Qed
MeasureTheory.LocallyIntegrableOn.continuousOn_mul ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : LocallyIntegrableOn f s hg : ContinuousOn g s hs : IsOpen s ⊢ LocallyIntegrableOn (fun x => g x * f x) s rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢ X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k hg : ContinuousOn g s hs : IsOpen s ⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => g x * f x) k exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c Qed
MeasureTheory.LocallyIntegrableOn.mul_continuousOn ** X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : LocallyIntegrableOn f s hg : ContinuousOn g s hs : IsOpen s ⊢ LocallyIntegrableOn (fun x => f x * g x) s rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢ X : Type u_1 Y : Type u_2 E : Type u_3 R : Type u_4 inst✝⁹ : MeasurableSpace X inst✝⁸ : TopologicalSpace X inst✝⁷ : MeasurableSpace Y inst✝⁶ : TopologicalSpace Y inst✝⁵ : NormedAddCommGroup E f✝ g✝ : X → E μ : Measure X s✝ : Set X inst✝⁴ : OpensMeasurableSpace X A K : Set X inst✝³ : LocallyCompactSpace X inst✝² : T2Space X inst✝¹ : NormedRing R inst✝ : SecondCountableTopologyEither X R f g : X → R s : Set X hf : ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k hg : ContinuousOn g s hs : IsOpen s ⊢ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn (fun x => f x * g x) k exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c Qed
MeasureTheory.SimpleFunc.nnnorm_approxOn_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ have := edist_approxOn_le hf h₀ x n α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ rw [edist_comm y₀] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist (↑(approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist (f x) y₀ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ simp only [edist_nndist, nndist_eq_nnnorm] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ↑‖f x - y₀‖₊ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ≤ ‖f x - y₀‖₊ exact_mod_cast this ** Qed
MeasureTheory.SimpleFunc.norm_approxOn_y₀_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ have := edist_approxOn_y0_le hf h₀ x n α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist y₀ (↑(approxOn f hf s y₀ h₀ n) x) ≤ edist y₀ (f x) + edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ repeat' rw [edist_comm y₀, edist_eq_coe_nnnorm_sub] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖₊ ≤ ↑‖f x - y₀‖₊ + ↑‖f x - y₀‖₊ ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ exact_mod_cast this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖₊ ≤ edist y₀ (f x) + edist y₀ (f x) ⊢ ‖↑(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ rw [edist_comm y₀, edist_eq_coe_nnnorm_sub] at this ** Qed
MeasureTheory.SimpleFunc.norm_approxOn_zero_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E h₀ : 0 ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ ⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖ have := edist_approxOn_y0_le hf h₀ x n α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E h₀ : 0 ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : edist 0 (↑(approxOn f hf s 0 h₀ n) x) ≤ edist 0 (f x) + edist 0 (f x) ⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖ simp [edist_comm (0 : E), edist_eq_coe_nnnorm] at this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E h₀ : 0 ∈ s inst✝ : SeparableSpace ↑s x : β n : ℕ this : ↑‖↑(approxOn f hf s 0 h₀ n) x‖₊ ≤ ↑‖f x‖₊ + ↑‖f x‖₊ ⊢ ‖↑(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖ exact_mod_cast this ** Qed
MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_snorm α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ n : ℕ ⊢ 0 ∈ Set.range f ∪ {0} simp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ Tendsto (fun n => snorm (↑(approxOn f fmeas (Set.range f ∪ {0}) 0 (_ : 0 ∈ Set.range f ∪ {0}) n) - f) p μ) atTop (𝓝 0) refine' tendsto_approxOn_Lp_snorm fmeas _ hp_ne_top _ _ case refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ ∀ᵐ (x : β) ∂μ, f x ∈ closure (Set.range f ∪ {0}) apply eventually_of_forall case refine'_1.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ ∀ (x : β), f x ∈ closure (Set.range f ∪ {0}) intro x case refine'_1.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ x : β ⊢ f x ∈ closure (Set.range f ∪ {0}) apply subset_closure case refine'_1.hp.a α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ x : β ⊢ f x ∈ Set.range f ∪ {0} simp case refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E hp_ne_top : p ≠ ⊤ μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : snorm f p μ < ⊤ ⊢ snorm (fun x => f x - 0) p μ < ⊤ simpa using hf ** Qed
MeasureTheory.SimpleFunc.memℒp_approxOn_range α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁵ : MeasurableSpace β inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : Memℒp f p n : ℕ ⊢ 0 ∈ Set.range f ∪ {0} simp ** Qed
MeasureTheory.Memℒp.exists_simpleFunc_snorm_sub_lt α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p borelize E α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p let f' := hf.1.mk f α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p rsuffices ⟨g, hg, g_mem⟩ : ∃ g : β →ₛ E, snorm (f' - ⇑g) p μ < ε ∧ Memℒp g p μ α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p have hf' : Memℒp f' p μ := hf.ae_eq hf.1.ae_eq_mk α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p have f'meas : Measurable f' := hf.1.measurable_mk α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p have : SeparableSpace (range f' ∪ {0} : Set E) := StronglyMeasurable.separableSpace_range_union_singleton hf.1.stronglyMeasurable_mk α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' this : SeparableSpace ↑(Set.range f' ∪ {0}) ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p rcases ((tendsto_approxOn_range_Lp_snorm hp_ne_top f'meas hf'.2).eventually <\| gt_mem_nhds hε.bot_lt).exists with ⟨n, hn⟩ case intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' this : SeparableSpace ↑(Set.range f' ∪ {0}) n : ℕ hn : snorm (↑(approxOn f' f'meas (Set.range f' ∪ {0}) 0 (_ : 0 ∈ Set.range f' ∪ {0}) n) - f') p μ < ε ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p rw [← snorm_neg, neg_sub] at hn case intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) hf' : Memℒp f' p f'meas : Measurable f' this : SeparableSpace ↑(Set.range f' ∪ {0}) n : ℕ hn : snorm (f' - ↑(approxOn f' f'meas (Set.range f' ∪ {0}) 0 (_ : 0 ∈ Set.range f' ∪ {0}) n)) p μ < ε ⊢ ∃ g, snorm (f' - ↑g) p μ < ε ∧ Memℒp (↑g) p exact ⟨_, hn, memℒp_approxOn_range f'meas hf' _⟩ case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ ∃ g, snorm (f - ↑g) p μ < ε ∧ Memℒp (↑g) p refine' ⟨g, _, g_mem⟩ case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ snorm (f - ↑g) p μ < ε suffices snorm (f - ⇑g) p μ = snorm (f' - ⇑g) p μ by rwa [this] case intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ snorm (f - ↑g) p μ = snorm (f' - ↑g) p μ apply snorm_congr_ae case intro.intro.hfg α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p ⊢ f - ↑g =ᵐ[μ] f' - ↑g filter_upwards [hf.1.ae_eq_mk] with x hx case h α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p x : β hx : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x ⊢ (f - ↑g) x = (f' - ↑g) x simpa only [Pi.sub_apply, sub_left_inj] using hx α : Type u_1 β : Type u_2 ι : Type u_3 E✝ : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E✝ inst✝² : NormedAddCommGroup E✝ inst✝¹ : NormedAddCommGroup F q : ℝ p : ℝ≥0∞ E : Type u_7 inst✝ : NormedAddCommGroup E f : β → E μ : Measure β hf : Memℒp f p hp_ne_top : p ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E f' : β → E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) g : β →ₛ E hg : snorm (f' - ↑g) p μ < ε g_mem : Memℒp (↑g) p this : snorm (f - ↑g) p μ = snorm (f' - ↑g) p μ ⊢ snorm (f - ↑g) p μ < ε rwa [this] ** Qed
MeasureTheory.SimpleFunc.tendsto_approxOn_L1_nnnorm α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s μ : Measure β hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s hi : HasFiniteIntegral fun x => f x - y₀ ⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f hf s y₀ h₀ n) x - f x‖₊ ∂μ) atTop (𝓝 0) simpa [snorm_one_eq_lintegral_nnnorm] using tendsto_approxOn_Lp_snorm hf h₀ one_ne_top hμ (by simpa [snorm_one_eq_lintegral_nnnorm] using hi) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E hf : Measurable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s μ : Measure β hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure s hi : HasFiniteIntegral fun x => f x - y₀ ⊢ snorm (fun x => f x - y₀) 1 μ < ⊤ simpa [snorm_one_eq_lintegral_nnnorm] using hi ** Qed
MeasureTheory.SimpleFunc.integrable_approxOn α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f hf : Integrable f s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s hi₀ : Integrable fun x => y₀ n : ℕ ⊢ Integrable ↑(approxOn f fmeas s y₀ h₀ n) rw [← memℒp_one_iff_integrable] at hf hi₀ ⊢ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f hf : Memℒp f 1 s : Set E y₀ : E h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s hi₀ : Memℒp (fun x => y₀) 1 n : ℕ ⊢ Memℒp (↑(approxOn f fmeas s y₀ h₀ n)) 1 exact memℒp_approxOn fmeas hf h₀ hi₀ n ** Qed
MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_nnnorm α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f n : ℕ x : β ⊢ 0 ∈ Set.range f ∪ {0} simp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ Tendsto (fun n => ∫⁻ (x : β), ↑‖↑(approxOn f fmeas (Set.range f ∪ {0}) 0 (_ : 0 ∈ Set.range f ∪ {0}) n) x - f x‖₊ ∂μ) atTop (𝓝 0) apply tendsto_approxOn_L1_nnnorm fmeas case hμ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ ∀ᵐ (x : β) ∂μ, f x ∈ closure (Set.range f ∪ {0}) apply eventually_of_forall case hμ.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ ∀ (x : β), f x ∈ closure (Set.range f ∪ {0}) intro x case hμ.hp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f x : β ⊢ f x ∈ closure (Set.range f ∪ {0}) apply subset_closure case hμ.hp.a α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f x : β ⊢ f x ∈ Set.range f ∪ {0} simp case hi α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : OpensMeasurableSpace E f : β → E μ : Measure β inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) fmeas : Measurable f hf : Integrable f ⊢ HasFiniteIntegral fun x => f x - 0 simpa using hf.2 ** Qed
MeasureTheory.SimpleFunc.integrable_approxOn_range α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace β inst✝³ : MeasurableSpace E inst✝² : NormedAddCommGroup E inst✝¹ : BorelSpace E f : β → E μ : Measure β fmeas : Measurable f inst✝ : SeparableSpace ↑(Set.range f ∪ {0}) hf : Integrable f n : ℕ ⊢ 0 ∈ Set.range f ∪ {0} simp ** Qed
MeasureTheory.SimpleFunc.snorm'_eq ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α ⊢ snorm' (↑f) p μ = (∑ y in SimpleFunc.range f, ↑‖y‖₊ ^ p * ↑↑μ (↑f ⁻¹' {y})) ^ (1 / p) have h_map : (fun a => (‖f a‖₊ : ℝ≥0∞) ^ p) = f.map fun a : F => (‖a‖₊ : ℝ≥0∞) ^ p := by simp; rfl α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α h_map : (fun a => ↑‖↑f a‖₊ ^ p) = ↑(map (fun a => ↑‖a‖₊ ^ p) f) ⊢ snorm' (↑f) p μ = (∑ y in SimpleFunc.range f, ↑‖y‖₊ ^ p * ↑↑μ (↑f ⁻¹' {y})) ^ (1 / p) rw [snorm', h_map, lintegral_eq_lintegral, map_lintegral] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α ⊢ (fun a => ↑‖↑f a‖₊ ^ p) = ↑(map (fun a => ↑‖a‖₊ ^ p) f) simp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F μ✝ : Measure α p✝ : ℝ≥0∞ p : ℝ f : α →ₛ F μ : Measure α ⊢ (fun a => ↑‖↑f a‖₊ ^ p) = (fun a => ↑‖a‖₊ ^ p) ∘ ↑f rfl Qed
MeasureTheory.SimpleFunc.measure_support_lt_top α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ⊢ ↑↑μ (support ↑f) < ⊤ rw [support_eq] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ ⊢ ↑↑μ (⋃ y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f), ↑f ⁻¹' {y}) < ⊤ refine' (measure_biUnion_finset_le _ _).trans_lt (ENNReal.sum_lt_top_iff.mpr fun y hy => _) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ y : β hy : y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f) ⊢ ↑↑μ (↑f ⁻¹' {y}) < ⊤ rw [Finset.mem_filter] at hy α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F μ : Measure α p : ℝ≥0∞ inst✝ : Zero β f : α →ₛ β hf : ∀ (y : β), y ≠ 0 → ↑↑μ (↑f ⁻¹' {y}) < ⊤ y : β hy : y ∈ SimpleFunc.range f ∧ y ≠ 0 ⊢ ↑↑μ (↑f ⁻¹' {y}) < ⊤ exact hf y hy.2 ** Qed
MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : { x // x ∈ simpleFunc E p μ } ⊢ ↑(toSimpleFunc f) =ᵐ[μ] ↑↑↑f convert (AEEqFun.coeFn_mk (toSimpleFunc f) (toSimpleFunc f).aestronglyMeasurable).symm using 2 case h.e'_5.h.e'_6 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : { x // x ∈ simpleFunc E p μ } ⊢ ↑↑f = AEEqFun.mk ↑(toSimpleFunc f) (_ : AEStronglyMeasurable (↑(toSimpleFunc f)) μ) exact (Classical.choose_spec f.2).symm ** Qed
MeasureTheory.Lp.simpleFunc.toSimpleFunc_toLp α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ ↑(toSimpleFunc (toLp f hfi)) =ᵐ[μ] ↑f rw [← AEEqFun.mk_eq_mk] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ AEEqFun.mk ↑(toSimpleFunc (toLp f hfi)) ?m.1694496 = AEEqFun.mk ↑f ?m.1694497 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ AEStronglyMeasurable (↑(toSimpleFunc (toLp f hfi))) μ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f : α →ₛ E hfi : Memℒp (↑f) p ⊢ AEStronglyMeasurable (↑f) μ exact Classical.choose_spec (toLp f hfi).2 ** Qed
MeasureTheory.Lp.simpleFunc.zero_toSimpleFunc α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α ⊢ ↑(toSimpleFunc 0) =ᵐ[μ] 0 filter_upwards [toSimpleFunc_eq_toFun (0 : Lp.simpleFunc E p μ), Lp.coeFn_zero E 1 μ] with _ h₁ _ case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α a✝¹ : α h₁ : ↑(toSimpleFunc 0) a✝¹ = ↑↑↑0 a✝¹ a✝ : ↑↑0 a✝¹ = OfNat.ofNat 0 a✝¹ ⊢ ↑(toSimpleFunc 0) a✝¹ = OfNat.ofNat 0 a✝¹ rwa [h₁] ** Qed
MeasureTheory.Lp.simpleFunc.add_toSimpleFunc α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } ⊢ ↑(toSimpleFunc (f + g)) =ᵐ[μ] ↑(toSimpleFunc f) + ↑(toSimpleFunc g) filter_upwards [toSimpleFunc_eq_toFun (f + g), toSimpleFunc_eq_toFun f, toSimpleFunc_eq_toFun g, Lp.coeFn_add (f : Lp E p μ) g] with _ case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑↑↑(f + g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑↑(↑f + ↑g) a✝ = (↑↑↑f + ↑↑↑g) a✝ → ↑(toSimpleFunc (f + g)) a✝ = (↑(toSimpleFunc f) + ↑(toSimpleFunc g)) a✝ simp only [AddSubgroup.coe_add, Pi.add_apply] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(toSimpleFunc (f + g)) a✝ = ↑(toSimpleFunc f) a✝ + ↑(toSimpleFunc g) a✝ iterate 4 intro h; rw [h] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝¹ : ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ h✝ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ ⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ intro h case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝² : ↑(toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝ h✝¹ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h✝ : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ h : ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ ⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ rw [h] ** Qed
MeasureTheory.Lp.simpleFunc.sub_toSimpleFunc α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } ⊢ ↑(toSimpleFunc (f - g)) =ᵐ[μ] ↑(toSimpleFunc f) - ↑(toSimpleFunc g) filter_upwards [toSimpleFunc_eq_toFun (f - g), toSimpleFunc_eq_toFun f, toSimpleFunc_eq_toFun g, Lp.coeFn_sub (f : Lp E p μ) g] with _ case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f - g)) a✝ = ↑↑↑(f - g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑↑(↑f - ↑g) a✝ = (↑↑↑f - ↑↑↑g) a✝ → ↑(toSimpleFunc (f - g)) a✝ = (↑(toSimpleFunc f) - ↑(toSimpleFunc g)) a✝ simp only [AddSubgroup.coe_sub, Pi.sub_apply] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α ⊢ ↑(toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝ → ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(toSimpleFunc (f - g)) a✝ = ↑(toSimpleFunc f) a✝ - ↑(toSimpleFunc g) a✝ repeat' intro h; rw [h] case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝¹ : ↑(toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝ h✝ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ ⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ intro h case h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α f g : { x // x ∈ simpleFunc E p μ } a✝ : α h✝² : ↑(toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝ h✝¹ : ↑(toSimpleFunc f) a✝ = ↑↑↑f a✝ h✝ : ↑(toSimpleFunc g) a✝ = ↑↑↑g a✝ h : ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ ⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ rw [h] ** Qed
MeasureTheory.Lp.simpleFunc.induction α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } ⊢ P f suffices ∀ f : α →ₛ E, ∀ hf : Memℒp f p μ, P (toLp f hf) by rw [← toLp_toSimpleFunc f] apply this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } ⊢ ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) clear f α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) ⊢ ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) apply SimpleFunc.induction α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } this : ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) ⊢ P f rw [← toLp_toSimpleFunc f] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f : { x // x ∈ simpleFunc E p μ } this : ∀ (f : α →ₛ E) (hf : Memℒp (↑f) p), P (toLp f hf) ⊢ P (toLp (toSimpleFunc f) (_ : Memℒp (↑(toSimpleFunc f)) p)) apply this case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) ⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p), P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) intro c s hs hf case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p ⊢ P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) by_cases hc : c = 0 case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : ¬c = 0 ⊢ P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) exact h_ind c hs (SimpleFunc.measure_lt_top_of_memℒp_indicator hp_pos hp_ne_top hc hs hf) case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) convert h_ind 0 MeasurableSet.empty (by simp) using 1 case h.e'_1 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf = indicatorConst p (_ : MeasurableSet ∅) (_ : ↑↑μ ∅ ≠ ⊤) 0 ext1 case h.e'_1.a α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ ↑(toLp (SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) hf) = ↑(indicatorConst p (_ : MeasurableSet ∅) (_ : ↑↑μ ∅ ≠ ⊤) 0) simp [hc] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) c : E s : Set α hs : MeasurableSet s hf : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ ↑↑μ ∅ < ⊤ simp case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) ⊢ ∀ ⦃f g : α →ₛ E⦄, Disjoint (support ↑f) (support ↑g) → (∀ (hf : Memℒp (↑f) p), P (toLp f hf)) → (∀ (hf : Memℒp (↑g) p), P (toLp g hf)) → ∀ (hf : Memℒp (↑(f + g)) p), P (toLp (f + g) hf) intro f g hfg hf hg hfg' case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : ∀ (hf : Memℒp (↑f) p), P (toLp f hf) hg : ∀ (hf : Memℒp (↑g) p), P (toLp g hf) hfg' : Memℒp (↑(f + g)) p ⊢ P (toLp (f + g) hfg') obtain ⟨hf', hg'⟩ : Memℒp f p μ ∧ Memℒp g p μ := (memℒp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable).mp hfg' case h_add.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α hp_pos : p ≠ 0 hp_ne_top : p ≠ ⊤ P : { x // x ∈ simpleFunc E p μ } → Prop h_ind : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : ↑↑μ s < ⊤), P (indicatorConst p hs (_ : ↑↑μ s ≠ ⊤) c) h_add : ∀ ⦃f g : α →ₛ E⦄ (hf : Memℒp (↑f) p) (hg : Memℒp (↑g) p), Disjoint (support ↑f) (support ↑g) → P (toLp f hf) → P (toLp g hg) → P (toLp f hf + toLp g hg) f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : ∀ (hf : Memℒp (↑f) p), P (toLp f hf) hg : ∀ (hf : Memℒp (↑g) p), P (toLp g hf) hfg' : Memℒp (↑(f + g)) p hf' : Memℒp (↑f) p hg' : Memℒp (↑g) p ⊢ P (toLp (f + g) hfg') exact h_add hf' hg' hfg (hf hf') (hg hg') ** Qed
MeasureTheory.Lp.simpleFunc.coeFn_le α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G f g : { x // x ∈ simpleFunc G p μ } ⊢ ↑↑↑f ≤ᵐ[μ] ↑↑↑g ↔ f ≤ g rw [← Subtype.coe_le_coe, ← Lp.coeFn_le] ** Qed
MeasureTheory.Lp.simpleFunc.exists_simpleFunc_nonneg_ae_eq α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G f : { x // x ∈ simpleFunc G p μ } hf : 0 ≤ f ⊢ ∃ f', 0 ≤ f' ∧ ↑↑↑f =ᵐ[μ] ↑f' rcases f with ⟨⟨f, hp⟩, g, (rfl : _ = f)⟩ case mk.mk.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G g : α →ₛ G hp : AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ) ∈ Lp G p hf : 0 ≤ { val := { val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }, property := (_ : ∃ s, AEEqFun.mk ↑s (_ : AEStronglyMeasurable (↑s) μ) = ↑{ val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }) } ⊢ ∃ f', 0 ≤ f' ∧ ↑↑↑{ val := { val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }, property := (_ : ∃ s, AEEqFun.mk ↑s (_ : AEStronglyMeasurable (↑s) μ) = ↑{ val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }) } =ᵐ[μ] ↑f' change 0 ≤ᵐ[μ] g at hf case mk.mk.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G g : α →ₛ G hp : AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ) ∈ Lp G p hf : 0 ≤ᵐ[μ] ↑g ⊢ ∃ f', 0 ≤ f' ∧ ↑↑↑{ val := { val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }, property := (_ : ∃ s, AEEqFun.mk ↑s (_ : AEStronglyMeasurable (↑s) μ) = ↑{ val := AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ), property := hp }) } =ᵐ[μ] ↑f' refine ⟨g ⊔ 0, le_sup_right, (AEEqFun.coeFn_mk _ _).trans ?_⟩ case mk.mk.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝³ : MeasurableSpace α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F p : ℝ≥0∞ μ : Measure α G : Type u_7 inst✝ : NormedLatticeAddCommGroup G g : α →ₛ G hp : AEEqFun.mk ↑g (_ : AEStronglyMeasurable (↑g) μ) ∈ Lp G p hf : 0 ≤ᵐ[μ] ↑g ⊢ ↑g =ᵐ[μ] ↑(g ⊔ 0) exact hf.mono fun x hx ↦ (sup_of_le_left hx).symm ** Qed
MeasureTheory.Memℒp.induction α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f ⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f have : ∀ f : Lp.simpleFunc E p μ, P f := by intro f exact h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memℒp f) (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memℒp f)) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this✝ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f this : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f ⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f have : ∀ f : Lp E p μ, P f := fun f => (Lp.simpleFunc.denseRange hp_ne_top).induction_on f h_closed this α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this✝¹ : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f this✝ : ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f this : ∀ (f : { x // x ∈ Lp E p }), P ↑↑f ⊢ ∀ ⦃f : α → E⦄, Memℒp f p → P f exact fun f hf => h_ae hf.coeFn_toLp (Lp.memℒp _) (this (hf.toLp f)) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g ⊢ ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f apply SimpleFunc.induction case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g ⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s), Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p → P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) intro c s hs h case h_ind α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) by_cases hc : c = 0 case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : ¬c = 0 ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) have hp_pos : p ≠ 0 := (lt_of_lt_of_le zero_lt_one _i.elim).ne' case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : ¬c = 0 hp_pos : p ≠ 0 ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) exact h_ind c hs (SimpleFunc.measure_lt_top_of_memℒp_indicator hp_pos hp_ne_top hc hs h) case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g c : E s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p hc : c = 0 ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) subst hc case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p ⊢ P ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) convert h_ind 0 MeasurableSet.empty (by simp) using 1 case h.e'_1 α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p ⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) = Set.indicator ∅ fun x => 0 ext case h.e'_1.h α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p x✝ : α ⊢ ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = Set.indicator ∅ (fun x => 0) x✝ simp [const] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g s : Set α hs : MeasurableSet s h : Memℒp (↑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p ⊢ ↑↑μ ∅ < ⊤ simp case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g ⊢ ∀ ⦃f g : α →ₛ E⦄, Disjoint (support ↑f) (support ↑g) → (Memℒp (↑f) p → P ↑f) → (Memℒp (↑g) p → P ↑g) → Memℒp (↑(f + g)) p → P ↑(f + g) intro f g hfg hf hg int_fg case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : Memℒp (↑f) p → P ↑f hg : Memℒp (↑g) p → P ↑g int_fg : Memℒp (↑(f + g)) p ⊢ P ↑(f + g) rw [SimpleFunc.coe_add, memℒp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable] at int_fg case h_add α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g f g : α →ₛ E hfg : Disjoint (support ↑f) (support ↑g) hf : Memℒp (↑f) p → P ↑f hg : Memℒp (↑g) p → P ↑g int_fg : Memℒp (↑f) p ∧ Memℒp (↑g) p ⊢ P ↑(f + g) refine' h_add hfg int_fg.1 int_fg.2 (hf int_fg.1) (hg int_fg.2) α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f ⊢ ∀ (f : { x // x ∈ Lp.simpleFunc E p μ }), P ↑↑↑f intro f α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α _i : Fact (1 ≤ p) hp_ne_top : p ≠ ⊤ P : (α → E) → Prop h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → ↑↑μ s < ⊤ → P (Set.indicator s fun x => c) h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Memℒp f p → Memℒp g p → P f → P g → P (f + g) h_closed : IsClosed {f \| P ↑↑f} h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Memℒp f p → P f → P g this : ∀ (f : α →ₛ E), Memℒp (↑f) p → P ↑f f : { x // x ∈ Lp.simpleFunc E p μ } ⊢ P ↑↑↑f exact h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.memℒp f) (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.memℒp f)) ** Qed
MeasureTheory.L1.SimpleFunc.integrable α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α f : { x // x ∈ Lp.simpleFunc E 1 μ } ⊢ Integrable ↑(Lp.simpleFunc.toSimpleFunc f) rw [← memℒp_one_iff_integrable] α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝¹ : MeasurableSpace α inst✝ : NormedAddCommGroup E f✝ : α → E p : ℝ≥0∞ μ : Measure α f : { x // x ∈ Lp.simpleFunc E 1 μ } ⊢ Memℒp (↑(Lp.simpleFunc.toSimpleFunc f)) 1 exact Lp.simpleFunc.memℒp f ** Qed
MeasureTheory.analyticSet_empty α : Type u_1 ι : Type u_2 inst✝ : TopologicalSpace α ⊢ AnalyticSet ∅ rw [AnalyticSet] α : Type u_1 ι : Type u_2 inst✝ : TopologicalSpace α ⊢ ∅ = ∅ ∨ ∃ f, Continuous f ∧ range f = ∅ exact Or.inl rfl ** Qed
IsOpen.analyticSet_image α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α β : Type u_3 inst✝¹ : TopologicalSpace β inst✝ : PolishSpace β s : Set β hs : IsOpen s f : β → α f_cont : Continuous f ⊢ AnalyticSet (f '' s) rw [image_eq_range] α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α β : Type u_3 inst✝¹ : TopologicalSpace β inst✝ : PolishSpace β s : Set β hs : IsOpen s f : β → α f_cont : Continuous f ⊢ AnalyticSet (range fun x => f ↑x) haveI : PolishSpace s := hs.polishSpace α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α β : Type u_3 inst✝¹ : TopologicalSpace β inst✝ : PolishSpace β s : Set β hs : IsOpen s f : β → α f_cont : Continuous f this : PolishSpace ↑s ⊢ AnalyticSet (range fun x => f ↑x) exact analyticSet_range_of_polishSpace (f_cont.comp continuous_subtype_val) ** Qed
MeasureTheory.AnalyticSet.image_of_continuousOn α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s ⊢ AnalyticSet (f '' s) rcases analyticSet_iff_exists_polishSpace_range.1 hs with ⟨γ, γtop, γpolish, g, g_cont, gs⟩ case intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s ⊢ AnalyticSet (f '' s) have : f '' s = range (f ∘ g) := by rw [range_comp, gs] case intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ AnalyticSet (f '' s) rw [this] case intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ AnalyticSet (range (f ∘ g)) apply analyticSet_range_of_polishSpace case intro.intro.intro.intro.intro.f_cont α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ Continuous (f ∘ g) apply hf.comp_continuous g_cont fun x => _ α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ ∀ (x : γ), g x ∈ s rw [← gs] α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s this : f '' s = range (f ∘ g) ⊢ ∀ (x : γ), g x ∈ range g exact mem_range_self α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α β : Type u_3 inst✝ : TopologicalSpace β s : Set α hs : AnalyticSet s f : α → β hf : ContinuousOn f s γ : Type γtop : TopologicalSpace γ γpolish : PolishSpace γ g : γ → α g_cont : Continuous g gs : range g = s ⊢ f '' s = range (f ∘ g) rw [range_comp, gs] ** Qed
MeasureTheory.AnalyticSet.iInter α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α hι : Nonempty ι inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) ⊢ AnalyticSet (⋂ n, s n) rcases hι with ⟨i₀⟩ case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι ⊢ AnalyticSet (⋂ n, s n) choose β hβ h'β f f_cont f_range using fun n => analyticSet_iff_exists_polishSpace_range.1 (hs n) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n ⊢ AnalyticSet (⋂ n, s n) let γ := ∀ n, β n case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n ⊢ AnalyticSet (⋂ n, s n) let t : Set γ := ⋂ n, { x \| f n (x n) = f i₀ (x i₀) } case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} ⊢ AnalyticSet (⋂ n, s n) have t_closed : IsClosed t := by apply isClosed_iInter intro n exact isClosed_eq ((f_cont n).comp (continuous_apply n)) ((f_cont i₀).comp (continuous_apply i₀)) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t ⊢ AnalyticSet (⋂ n, s n) haveI : PolishSpace t := t_closed.polishSpace case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t ⊢ AnalyticSet (⋂ n, s n) let F : t → α := fun x => f i₀ ((x : γ) i₀) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) ⊢ AnalyticSet (⋂ n, s n) have F_cont : Continuous F := (f_cont i₀).comp ((continuous_apply i₀).comp continuous_subtype_val) case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F F_range : range F = ⋂ n, s n ⊢ AnalyticSet (⋂ n, s n) rw [← F_range] case intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F F_range : range F = ⋂ n, s n ⊢ AnalyticSet (range F) exact analyticSet_range_of_polishSpace F_cont α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} ⊢ IsClosed t apply isClosed_iInter case h α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} ⊢ ∀ (i : ι), IsClosed {x \| f i (x i) = f i₀ (x i₀)} intro n case h α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} n : ι ⊢ IsClosed {x \| f n (x n) = f i₀ (x i₀)} exact isClosed_eq ((f_cont n).comp (continuous_apply n)) ((f_cont i₀).comp (continuous_apply i₀)) α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F ⊢ range F = ⋂ n, s n apply Subset.antisymm case h₁ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F ⊢ range F ⊆ ⋂ n, s n rintro y ⟨x, rfl⟩ case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t ⊢ F x ∈ ⋂ n, s n refine mem_iInter.2 fun n => ?_ case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t n : ι ⊢ F x ∈ s n have : f n ((x : γ) n) = F x := (mem_iInter.1 x.2 n : _) case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this✝ : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t n : ι this : f n (↑x n) = F x ⊢ F x ∈ s n rw [← this, ← f_range n] case h₁.intro α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this✝ : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F x : ↑t n : ι this : f n (↑x n) = F x ⊢ f n (↑x n) ∈ range (f n) exact mem_range_self _ case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F ⊢ ⋂ n, s n ⊆ range F intro y hy case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n ⊢ y ∈ range F have A : ∀ n, ∃ x : β n, f n x = y := by intro n rw [← mem_range, f_range n] exact mem_iInter.1 hy n case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n A : ∀ (n : ι), ∃ x, f n x = y ⊢ y ∈ range F choose x hx using A case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y ⊢ y ∈ range F have xt : x ∈ t := by refine mem_iInter.2 fun n => ?_ simp [hx] case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y xt : x ∈ t ⊢ y ∈ range F refine' ⟨⟨x, xt⟩, _⟩ case h₂ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y xt : x ∈ t ⊢ F { val := x, property := xt } = y exact hx i₀ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n ⊢ ∀ (n : ι), ∃ x, f n x = y intro n α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n n : ι ⊢ ∃ x, f n x = y rw [← mem_range, f_range n] α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n n : ι ⊢ y ∈ s n exact mem_iInter.1 hy n α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y ⊢ x ∈ t refine mem_iInter.2 fun n => ?_ α : Type u_1 ι : Type u_2 inst✝² : TopologicalSpace α inst✝¹ : Countable ι inst✝ : T2Space α s : ι → Set α hs : ∀ (n : ι), AnalyticSet (s n) i₀ : ι β : ι → Type hβ : (n : ι) → TopologicalSpace (β n) h'β : ∀ (n : ι), PolishSpace (β n) f : (n : ι) → β n → α f_cont : ∀ (n : ι), Continuous (f n) f_range : ∀ (n : ι), range (f n) = s n γ : Type u_2 := (n : ι) → β n t : Set γ := ⋂ n, {x \| f n (x n) = f i₀ (x i₀)} t_closed : IsClosed t this : PolishSpace ↑t F : ↑t → α := fun x => f i₀ (↑x i₀) F_cont : Continuous F y : α hy : y ∈ ⋂ n, s n x : (n : ι) → β n hx : ∀ (n : ι), f n (x n) = y n : ι ⊢ x ∈ {x \| f n (x n) = f i₀ (x i₀)} simp [hx] ** Qed
IsClosed.analyticSet α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : PolishSpace α s : Set α hs : IsClosed s ⊢ AnalyticSet s haveI : PolishSpace s := hs.polishSpace α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : PolishSpace α s : Set α hs : IsClosed s this : PolishSpace ↑s ⊢ AnalyticSet s rw [← @Subtype.range_val α s] α : Type u_1 ι : Type u_2 inst✝¹ : TopologicalSpace α inst✝ : PolishSpace α s : Set α hs : IsClosed s this : PolishSpace ↑s ⊢ AnalyticSet (range Subtype.val) exact analyticSet_range_of_polishSpace continuous_subtype_val ** Qed
MeasurableSet.analyticSet_image α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f ⊢ AnalyticSet (f '' s) letI := upgradeStandardBorel X α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X ⊢ AnalyticSet (f '' s) rcases hf.exists_continuous with ⟨τ', hle, hfc, hτ'⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X ⊢ AnalyticSet (f '' s) letI m' : MeasurableSpace X := @borel _ τ' case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X m' : MeasurableSpace X := borel X ⊢ AnalyticSet (f '' s) haveI b' : BorelSpace X := ⟨rfl⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X m' : MeasurableSpace X := borel X b' : BorelSpace X ⊢ AnalyticSet (f '' s) have hle := borel_anti hle case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝⁶ : TopologicalSpace α X : Type u_3 Y : Type u_4 inst✝⁵ : MeasurableSpace X inst✝⁴ : StandardBorelSpace X inst✝³ : TopologicalSpace Y inst✝² : MeasurableSpace Y inst✝¹ : OpensMeasurableSpace Y f : X → Y inst✝ : SecondCountableTopology ↑(range f) s : Set X hs : MeasurableSet s hf : Measurable f this : UpgradedStandardBorel X := upgradeStandardBorel X τ' : TopologicalSpace X hle✝ : τ' ≤ UpgradedStandardBorel.toTopologicalSpace hfc : Continuous f hτ' : PolishSpace X m' : MeasurableSpace X := borel X b' : BorelSpace X hle : borel X ≤ borel X ⊢ AnalyticSet (f '' s) exact (hle _ hs).analyticSet.image_of_continuous hfc ** Qed
MeasureTheory.AnalyticSet.measurableSet_of_compl α : Type u_1 ι : Type u_2 inst✝³ : TopologicalSpace α inst✝² : T2Space α inst✝¹ : MeasurableSpace α inst✝ : OpensMeasurableSpace α s : Set α hs : AnalyticSet s hsc : AnalyticSet sᶜ ⊢ MeasurableSet s rcases hs.measurablySeparable hsc disjoint_compl_right with ⟨u, hsu, hdu, hmu⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝³ : TopologicalSpace α inst✝² : T2Space α inst✝¹ : MeasurableSpace α inst✝ : OpensMeasurableSpace α s : Set α hs : AnalyticSet s hsc : AnalyticSet sᶜ u : Set α hsu : s ⊆ u hdu : Disjoint sᶜ u hmu : MeasurableSet u ⊢ MeasurableSet s obtain rfl : s = u := hsu.antisymm (disjoint_compl_left_iff_subset.1 hdu) case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝³ : TopologicalSpace α inst✝² : T2Space α inst✝¹ : MeasurableSpace α inst✝ : OpensMeasurableSpace α s : Set α hs : AnalyticSet s hsc : AnalyticSet sᶜ hsu : s ⊆ s hdu : Disjoint sᶜ s hmu : MeasurableSet s ⊢ MeasurableSet s exact hmu ** Qed
Measurable.map_measurableSpace_eq_borel α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f ⊢ MeasurableSpace.map f inst✝⁷ = borel Y have d := hf.mono le_rfl OpensMeasurableSpace.borel_le α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f d : Measurable f ⊢ MeasurableSpace.map f inst✝⁷ = borel Y letI := borel Y α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f d : Measurable f this : MeasurableSpace Y := borel Y ⊢ MeasurableSpace.map f inst✝⁷ = borel Y haveI : BorelSpace Y := ⟨rfl⟩ α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β inst✝ : SecondCountableTopology Y f : X → Y hf : Measurable f hsurj : Surjective f d : Measurable f this✝ : MeasurableSpace Y := borel Y this : BorelSpace Y ⊢ MeasurableSpace.map f inst✝⁷ = borel Y exact d.map_measurableSpace_eq hsurj ** Qed
Measurable.measurableSet_preimage_iff_inter_range α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 β : Type u_5 inst✝⁷ : MeasurableSpace X inst✝⁶ : StandardBorelSpace X inst✝⁵ : TopologicalSpace Y inst✝⁴ : T2Space Y inst✝³ : MeasurableSpace Y inst✝² : OpensMeasurableSpace Y inst✝¹ : MeasurableSpace β f : X → Y inst✝ : SecondCountableTopology ↑(range f) hf : Measurable f hr : MeasurableSet (range f) s : Set Y ⊢ MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f) rw [hf.measurableSet_preimage_iff_preimage_val, ← (MeasurableEmbedding.subtype_coe hr).measurableSet_image, Subtype.image_preimage_coe] ** Qed
Continuous.map_eq_borel α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : PolishSpace X inst✝⁴ : MeasurableSpace X inst✝³ : BorelSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f ⊢ MeasurableSpace.map f inst✝⁴ = borel Y borelize Y α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁶ : TopologicalSpace X inst✝⁵ : PolishSpace X inst✝⁴ : MeasurableSpace X inst✝³ : BorelSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f this✝¹ : MeasurableSpace Y := borel Y this✝ : BorelSpace Y ⊢ MeasurableSpace.map f inst✝⁴ = borel Y exact hf.measurable.map_measurableSpace_eq hsurj ** Qed
Continuous.map_borel_eq α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁴ : TopologicalSpace X inst✝³ : PolishSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f ⊢ MeasurableSpace.map f (borel X) = borel Y borelize X α : Type u_1 ι : Type u_2 X : Type u_3 Y : Type u_4 inst✝⁴ : TopologicalSpace X inst✝³ : PolishSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y inst✝ : SecondCountableTopology Y f : X → Y hf : Continuous f hsurj : Surjective f this✝¹ : MeasurableSpace X := borel X this✝ : BorelSpace X ⊢ MeasurableSpace.map f (borel X) = borel Y exact hf.map_eq_borel hsurj ** Qed
ContinuousOn.measurableEmbedding α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s ⊢ ∀ ⦃s_1 : Set ↑s⦄, MeasurableSet s_1 → MeasurableSet (restrict s f '' s_1) intro u hu α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s u : Set ↑s hu : MeasurableSet u ⊢ MeasurableSet (restrict s f '' u) have A : MeasurableSet (((↑) : s → γ) '' u) := (MeasurableEmbedding.subtype_coe hs).measurableSet_image.2 hu α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s u : Set ↑s hu : MeasurableSet u A : MeasurableSet (Subtype.val '' u) ⊢ MeasurableSet (restrict s f '' u) have B : MeasurableSet (f '' (((↑) : s → γ) '' u)) := A.image_of_continuousOn_injOn (f_cont.mono (Subtype.coe_image_subset s u)) (f_inj.mono (Subtype.coe_image_subset s u)) α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁶ : T2Space β inst✝⁵ : MeasurableSpace β s : Set γ f : γ → β inst✝⁴ : BorelSpace β inst✝³ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : MeasurableSet s f_cont : ContinuousOn f s f_inj : InjOn f s u : Set ↑s hu : MeasurableSet u A : MeasurableSet (Subtype.val '' u) B : MeasurableSet (f '' (Subtype.val '' u)) ⊢ MeasurableSet (restrict s f '' u) rwa [← image_comp] at B ** Qed
MeasureTheory.isClopenable_iff_measurableSet α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁴ : T2Space β inst✝³ : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ ⊢ IsClopenable s ↔ MeasurableSet s refine' ⟨fun hs => _, fun hs => hs.isClopenable⟩ α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝⁴ : T2Space β inst✝³ : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝² : PolishSpace γ inst✝¹ : MeasurableSpace γ inst✝ : BorelSpace γ hs : IsClopenable s ⊢ MeasurableSet s borelize γ α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ hs : IsClopenable s inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ ⊢ MeasurableSet s obtain ⟨t', t't, t'_polish, _, s_open⟩ : ∃ t' : TopologicalSpace γ, t' ≤ tγ ∧ @PolishSpace γ t' ∧ IsClosed[t'] s ∧ IsOpen[t'] s := hs case intro.intro.intro.intro α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ t' : TopologicalSpace γ t't : t' ≤ tγ t'_polish : PolishSpace γ left✝ : IsClosed s s_open : IsOpen s ⊢ MeasurableSet s rw [← borel_eq_borel_of_le t'_polish _ t't] α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ t' : TopologicalSpace γ t't : t' ≤ tγ t'_polish : PolishSpace γ left✝ : IsClosed s s_open : IsOpen s ⊢ PolishSpace γ infer_instance case intro.intro.intro.intro α : Type u_1 ι : Type u_2 γ : Type u_3 β : Type u_4 tβ : TopologicalSpace β inst✝³ : T2Space β inst✝² : MeasurableSpace β s : Set γ f : γ → β tγ : TopologicalSpace γ inst✝¹ : PolishSpace γ inst✝ : BorelSpace γ this✝ : MeasurableSpace γ := borel γ t' : TopologicalSpace γ t't : t' ≤ tγ t'_polish : PolishSpace γ left✝ : IsClosed s s_open : IsOpen s ⊢ MeasurableSet s exact MeasurableSpace.measurableSet_generateFrom s_open ** Qed
MeasurableSet.isClopenable' α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s ⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s letI := upgradeStandardBorel α α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α ⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s obtain ⟨t, hle, ht, s_clopen⟩ := hs.isClopenable case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s refine' ⟨t, _, ht, s_clopen⟩ case intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ BorelSpace α constructor case intro.intro.intro.measurable_eq α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ inst✝¹ = borel α rw [eq_borel_upgradeStandardBorel α, borel_eq_borel_of_le ht _ hle] α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s this : UpgradedStandardBorel α := upgradeStandardBorel α t : TopologicalSpace α hle : t ≤ UpgradedStandardBorel.toTopologicalSpace ht : PolishSpace α s_clopen : IsClosed s ∧ IsOpen s ⊢ PolishSpace α infer_instance ** Qed
MeasurableSet.standardBorel α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s ⊢ StandardBorelSpace ↑s obtain ⟨_, _, _, s_closed, _⟩ := hs.isClopenable' case intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s w✝ : TopologicalSpace α left✝¹ : BorelSpace α left✝ : PolishSpace α s_closed : IsClosed s right✝ : IsOpen s ⊢ StandardBorelSpace ↑s haveI := s_closed.polishSpace case intro.intro.intro.intro α : Type u_1 ι : Type u_2 inst✝¹ : MeasurableSpace α inst✝ : StandardBorelSpace α s : Set α hs : MeasurableSet s w✝ : TopologicalSpace α left✝¹ : BorelSpace α left✝ : PolishSpace α s_closed : IsClosed s right✝ : IsOpen s this : PolishSpace ↑s ⊢ StandardBorelSpace ↑s infer_instance ** Qed
MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite α : Type u_1 ι : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : StandardBorelSpace α inst✝ : Finite α ⊢ ∃ n, Nonempty (α ≃ᵐ ↑(range fun x => ↑↑x)) obtain ⟨n, ⟨n_equiv⟩⟩ := Finite.exists_equiv_fin α case intro.intro α : Type u_1 ι : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : StandardBorelSpace α inst✝ : Finite α n : ℕ n_equiv : α ≃ Fin n ⊢ ∃ n, Nonempty (α ≃ᵐ ↑(range fun x => ↑↑x)) refine' ⟨n, ⟨PolishSpace.Equiv.measurableEquiv (n_equiv.trans _)⟩⟩ case intro.intro α : Type u_1 ι : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : StandardBorelSpace α inst✝ : Finite α n : ℕ n_equiv : α ≃ Fin n ⊢ Fin n ≃ ↑(range fun x => ↑↑x) exact Equiv.ofInjective _ (Nat.cast_injective.comp Fin.val_injective) ** Qed
MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α ⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast)) have : PolishSpace (range ((↑) : ℕ → ℝ)) := Nat.closedEmbedding_coe_real.isClosedMap.closed_range.polishSpace α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α this : PolishSpace ↑(range Nat.cast) ⊢ Nonempty (α ≃ᵐ ↑(range Nat.cast)) refine' ⟨PolishSpace.Equiv.measurableEquiv _⟩ α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α this : PolishSpace ↑(range Nat.cast) ⊢ α ≃ ↑(range Nat.cast) refine' (nonempty_equiv_of_countable.some : α ≃ ℕ).trans _ α : Type u_1 ι : Type u_2 inst✝³ : MeasurableSpace α inst✝² : StandardBorelSpace α inst✝¹ : Infinite α inst✝ : Countable α this : PolishSpace ↑(range Nat.cast) ⊢ ℕ ≃ ↑(range Nat.cast) exact Equiv.ofInjective ((↑) : ℕ → ℝ) Nat.cast_injective ** Qed
Besicovitch.SatelliteConfig.inter' α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) rcases lt_or_le i (last N) with (H \| H) case inl α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : i < last N ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) exact a.inter i H case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : last N ≤ i ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) have I : i = last N := top_le_iff.1 H case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : last N ≤ i I : i = last N ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) have := (a.rpos (last N)).le case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) H : last N ≤ i I : i = last N this : 0 ≤ r a (last N) ⊢ dist (c a i) (c a (last N)) ≤ r a i + r a (last N) simp only [I, add_nonneg this this, dist_self] ** Qed
Besicovitch.SatelliteConfig.hlast' ** α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ ⊢ r a (last N) ≤ τ * r a i rcases lt_or_le i (last N) with (H \| H) case inl α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : i < last N ⊢ r a (last N) ≤ τ * r a i exact (a.hlast i H).2 case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : last N ≤ i ⊢ r a (last N) ≤ τ * r a i have : i = last N := top_le_iff.1 H case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : last N ≤ i this : i = last N ⊢ r a (last N) ≤ τ * r a i rw [this] case inr α : Type u_1 inst✝ : MetricSpace α N : ℕ τ : ℝ a : SatelliteConfig α N τ i : Fin (Nat.succ N) h : 1 ≤ τ H : last N ≤ i this : i = last N ⊢ r a (last N) ≤ τ * r a (last N) exact le_mul_of_one_le_left (a.rpos _).le h Qed
Besicovitch.TauPackage.monotone_iUnionUpTo α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α ⊢ Monotone (iUnionUpTo p) intro i j hij α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i j : Ordinal.{u} hij : i ≤ j ⊢ iUnionUpTo p i ≤ iUnionUpTo p j simp only [iUnionUpTo] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i j : Ordinal.{u} hij : i ≤ j ⊢ ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ≤ ⋃ j_1, ball (BallPackage.c p.toBallPackage (index p ↑j_1)) (BallPackage.r p.toBallPackage (index p ↑j_1)) exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩ ** Qed
Besicovitch.TauPackage.lastStep_nonempty ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α ⊢ Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} by_contra h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ False suffices H : Function.Injective p.index α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} H : Function.Injective (index p) ⊢ False case H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ Function.Injective (index p) exact not_injective_of_ordinal p.index H case H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ Function.Injective (index p) intro x y hxy case H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y ⊢ x = y wlog x_le_y : x ≤ y generalizing x y α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y ⊢ x = y rcases eq_or_lt_of_le x_le_y with (rfl \| H) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y ⊢ x = y simp only [nonempty_def, not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] at h case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∀ (x : Ordinal.{u}), ∃ x_1, ¬BallPackage.c p.toBallPackage x_1 ∈ iUnionUpTo p x ∧ R p x ≤ p.τ * BallPackage.r p.toBallPackage x_1 ⊢ x = y specialize h y case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ x = y ** have A : p.c (p.index y) ∉ p.iUnionUpTo y := by have : p.index y = Classical.epsilon fun b : β => p.c b ∉ p.iUnionUpTo y ∧ p.R y ≤ p.τ * p.r b := by rw [TauPackage.index]; rfl rw [this] exact (Classical.epsilon_spec h).1 ** case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : ¬BallPackage.c p.toBallPackage (index p y) ∈ iUnionUpTo p y ⊢ x = y simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, Subtype.exists, Subtype.coe_mk] at A case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : ∀ (x : Ordinal.{u}), x < y → ¬BallPackage.c p.toBallPackage (index p y) ∈ ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ⊢ x = y specialize A x H case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : ¬BallPackage.c p.toBallPackage (index p y) ∈ ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ⊢ x = y simp [hxy] at A case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x A : BallPackage.r p.toBallPackage (index p y) ≤ 0 ⊢ x = y exact (lt_irrefl _ ((p.rpos (p.index y)).trans_le A)).elim case H.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x y : Ordinal.{u} hxy : index p x = index p y this : ∀ ⦃x y : Ordinal.{u}⦄, index p x = index p y → x ≤ y → x = y x_le_y : ¬x ≤ y ⊢ x = y exact (this hxy.symm (le_of_not_le x_le_y)).symm case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α h : ¬Set.Nonempty {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} x : Ordinal.{u} hxy : index p x = index p x x_le_y : x ≤ x ⊢ x = x rfl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ ¬BallPackage.c p.toBallPackage (index p y) ∈ iUnionUpTo p y ** have : p.index y = Classical.epsilon fun b : β => p.c b ∉ p.iUnionUpTo y ∧ p.R y ≤ p.τ * p.r b := by rw [TauPackage.index]; rfl ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x this : index p y = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b ⊢ ¬BallPackage.c p.toBallPackage (index p y) ∈ iUnionUpTo p y rw [this] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x this : index p y = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b ⊢ ¬BallPackage.c p.toBallPackage (Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b) ∈ iUnionUpTo p y exact (Classical.epsilon_spec h).1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ index p y = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b rw [TauPackage.index] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x y : Ordinal.{u} hxy : index p x = index p y x_le_y : x ≤ y H : x < y h : ∃ x, ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage x ⊢ (Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∧ ⨆ b, BallPackage.r p.toBallPackage ↑b ≤ p.τ * BallPackage.r p.toBallPackage b) = Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p y ∧ R p y ≤ p.τ * BallPackage.r p.toBallPackage b rfl Qed
Besicovitch.TauPackage.mem_iUnionUpTo_lastStep α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β ⊢ BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) have A : ∀ z : β, p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep := by have : p.lastStep ∈ {i \| ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} := csInf_mem p.lastStep_nonempty simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem] ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) ⊢ BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) by_contra h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) ⊢ False rcases A x with (H \| H) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) ⊢ False have Rpos : 0 < p.R p.lastStep := by apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) ⊢ False ** have B : p.τ⁻¹ * p.R p.lastStep < p.R p.lastStep := by conv_rhs => rw [← one_mul (p.R p.lastStep)] exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one ** case inr.intro.intro α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y ⊢ False rcases A y with (Hy \| Hy) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β ⊢ ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) ** have : p.lastStep ∈ {i \| ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b} := csInf_mem p.lastStep_nonempty ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β this : lastStep p ∈ {i \| ¬∃ b, ¬BallPackage.c p.toBallPackage b ∈ iUnionUpTo p i ∧ R p i ≤ p.τ * BallPackage.r p.toBallPackage b} ⊢ ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem] case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) ⊢ False exact h H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) ⊢ 0 < R p (lastStep p) apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) ⊢ p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) conv_rhs => rw [← one_mul (p.R p.lastStep)] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) ⊢ p.τ⁻¹ * R p (lastStep p) < 1 * R p (lastStep p) exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) ⊢ ∃ y, ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) ∧ p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y have := exists_lt_of_lt_csSup ?_ B case refine_1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) ⊢ Set.Nonempty (range fun b => BallPackage.r p.toBallPackage ↑b) rw [← image_univ, nonempty_image_iff] case refine_1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) ⊢ Set.Nonempty univ exact ⟨⟨_, h⟩, mem_univ _⟩ case refine_2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) this : ∃ a, (a ∈ range fun b => BallPackage.r p.toBallPackage ↑b) ∧ p.τ⁻¹ * R p (lastStep p) < a ⊢ ∃ y, ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) ∧ p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists, Subtype.coe_mk] case inr.intro.intro.inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y Hy : BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) ⊢ False exact hy1 Hy case inr.intro.intro.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : p.τ⁻¹ * R p (lastStep p) < BallPackage.r p.toBallPackage y Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p) ⊢ False rw [← div_eq_inv_mul] at hy2 case inr.intro.intro.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : R p (lastStep p) / p.τ < BallPackage.r p.toBallPackage y Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p) ⊢ False have := (div_le_iff' (_root_.zero_lt_one.trans p.one_lt_tau)).1 hy2.le case inr.intro.intro.inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α x : β A : ∀ (z : β), BallPackage.c p.toBallPackage z ∈ iUnionUpTo p (lastStep p) ∨ p.τ * BallPackage.r p.toBallPackage z < R p (lastStep p) h : ¬BallPackage.c p.toBallPackage x ∈ iUnionUpTo p (lastStep p) H : p.τ * BallPackage.r p.toBallPackage x < R p (lastStep p) Rpos : 0 < R p (lastStep p) B : p.τ⁻¹ * R p (lastStep p) < R p (lastStep p) y : β hy1 : ¬BallPackage.c p.toBallPackage y ∈ iUnionUpTo p (lastStep p) hy2 : R p (lastStep p) / p.τ < BallPackage.r p.toBallPackage y Hy : p.τ * BallPackage.r p.toBallPackage y < R p (lastStep p) this : R p (lastStep p) ≤ p.τ * BallPackage.r p.toBallPackage y ⊢ False exact lt_irrefl _ (Hy.trans_le this) Qed
Besicovitch.TauPackage.color_lt α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i : Ordinal.{u} hi : i < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) ⊢ color p i < N induction' i using Ordinal.induction with i IH case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p ⊢ color p i < N let A : Set ℕ := ⋃ (j : { j // j < i }) (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty), {p.color j} case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} ⊢ color p i < N have color_i : p.color i = sInf (univ \ A) := by rw [color] case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ color p i < N rw [color_i] case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ sInf (univ \ A) < N have N_mem : N ∈ univ \ A := by simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] intro j ji _ exact (IH j ji (ji.trans hi)).ne' case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A ⊢ sInf (univ \ A) ≠ N intro Inf_eq_N case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N ⊢ False have : ∀ k, k < N → ∃ j, j < i ∧ (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧ k = p.color j := by intro k hk rw [← Inf_eq_N] at hk have : k ∈ A := by simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk simp [and_assoc, -exists_and_left] at this simpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists, Subtype.coe_mk] case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N this : ∀ (k : ℕ), k < N → ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j ⊢ False choose! g hg using this case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) ⊢ False let G : ℕ → Ordinal := fun n => if n = N then i else g n case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p ⊢ False have fGn : ∀ n, n ≤ N → p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n)) := by intro n hn have : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by rw [index]; rfl rw [this] have : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _) exact Classical.epsilon_spec this ** case h α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) sc : SatelliteConfig α N p.τ := { c := fun k => BallPackage.c p.toBallPackage (index p (G ↑k)), r := fun k => BallPackage.r p.toBallPackage (index p (G ↑k)), rpos := (_ : ∀ (k : Fin (Nat.succ N)), 0 < BallPackage.r p.toBallPackage (index p (G ↑k))), h := (_ : ∀ (a b : Fin (Nat.succ N)), a ≠ b → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b), hlast := (_ : ∀ (a : Fin (N + 1)), a < last N → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑(last N)))) ∧ BallPackage.r p.toBallPackage (index p (G ↑(last N))) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a))), inter := (_ : ∀ (a : Fin (N + 1)), a < last N → dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N)) } ⊢ False exact hN.false sc α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} ⊢ color p i = sInf (univ \ A) rw [color] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ N ∈ univ \ A simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) ⊢ ∀ (x : Ordinal.{u}), x < i → Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) → ¬N = color p x intro j ji _ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) j : Ordinal.{u} ji : j < i a✝ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ⊢ ¬N = color p j exact (IH j ji (ji.trans hi)).ne' α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A this : sInf (univ \ A) ≠ N ⊢ sInf (univ \ A) < N rcases (csInf_le (OrderBot.bddBelow (univ \ A)) N_mem).lt_or_eq with (H \| H) case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A this : sInf (univ \ A) ≠ N H : sInf (univ \ A) < N ⊢ sInf (univ \ A) < N exact H case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A this : sInf (univ \ A) ≠ N H : sInf (univ \ A) = N ⊢ sInf (univ \ A) < N exact (this H).elim α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N ⊢ ∀ (k : ℕ), k < N → ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j intro k hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < N ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j rw [← Inf_eq_N] at hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j have : k ∈ A := by simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) this : k ∈ A ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j simp [and_assoc, -exists_and_left] at this α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) this : ∃ a, a < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p a)) (BallPackage.r p.toBallPackage (index p a)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p a ⊢ ∃ j, j < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p j simpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists, Subtype.coe_mk] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N k : ℕ hk : k < sInf (univ \ A) ⊢ k ∈ A simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n ⊢ ∀ (n : ℕ), n ≤ N → color p (G n) = n intro n hn α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n n : ℕ hn : n ≤ N ⊢ color p (G n) = n rcases hn.eq_or_lt with (rfl \| H) case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 hn : n ≤ n ⊢ color p (G n) = n simp only case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 hn : n ≤ n ⊢ color p (if True then i else g n) = n simp only [color_i, Inf_eq_N, if_true, eq_self_iff_true] case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n n : ℕ hn : n ≤ N H : n < N ⊢ color p (G n) = n simp only case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n n : ℕ hn : n ≤ N H : n < N ⊢ color p (if n = N then i else g n) = n simp only [H.ne, (hg n H).right.right.symm, if_false] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n ⊢ ∀ (n : ℕ), n ≤ N → G n < lastStep p intro n hn α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n n : ℕ hn : n ≤ N ⊢ G n < lastStep p rcases hn.eq_or_lt with (rfl \| H) case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 color_G : ∀ (n_1 : ℕ), n_1 ≤ n → color p (G n_1) = n_1 hn : n ≤ n ⊢ G n < lastStep p simp only case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p i : Ordinal.{u} hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) g : ℕ → Ordinal.{u} n : ℕ hN : IsEmpty (SatelliteConfig α n p.τ) IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < n N_mem : n ∈ univ \ A Inf_eq_N : sInf (univ \ A) = n hg : ∀ (k : ℕ), k < n → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1 color_G : ∀ (n_1 : ℕ), n_1 ≤ n → color p (G n_1) = n_1 hn : n ≤ n ⊢ (if True then i else g n) < lastStep p simp only [hi, if_true, eq_self_iff_true] case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n n : ℕ hn : n ≤ N H : n < N ⊢ G n < lastStep p simp only case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n n : ℕ hn : n ≤ N H : n < N ⊢ (if n = N then i else g n) < lastStep p simp only [H.ne, (hg n H).left.trans hi, if_false] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p ⊢ ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) intro n hn α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N ⊢ ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) ** have : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by rw [index]; rfl ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) rw [this] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ¬BallPackage.c p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) ** have : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t := by simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _) ** α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this✝ : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t this : ∃ t, ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ¬BallPackage.c p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t) exact Classical.epsilon_spec this α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N ⊢ index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t rw [index] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N ⊢ (Classical.epsilon fun b => ¬BallPackage.c p.toBallPackage b ∈ ⋃ j, ball (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∧ ⨆ b, BallPackage.r p.toBallPackage ↑b ≤ p.τ * BallPackage.r p.toBallPackage b) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t rfl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p n : ℕ hn : n ≤ N this : index p (G n) = Classical.epsilon fun t => ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t ⊢ ∃ t, ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage t simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) ⊢ ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) intro a b G_lt α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) have ha : (a : ℕ) ≤ N := Nat.lt_succ_iff.1 a.2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) have hb : (b : ℕ) ≤ N := Nat.lt_succ_iff.1 b.2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) constructor case left α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) have := (fGn b hb).1 case left α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N this : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑b) ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, Subtype.exists, Subtype.coe_mk] at this case left α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N this : ∀ (x : Ordinal.{u}), (x < if ↑b = N then i else g ↑b) → ¬BallPackage.c p.toBallPackage (index p (if ↑b = N then i else g ↑b)) ∈ ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) ⊢ BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) simpa only [dist_comm, mem_ball, not_lt] using this (G a) G_lt case right α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) apply le_trans _ (fGn a ha).2 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ R p (G ↑a) have B : p.c (p.index (G b)) ∉ p.iUnionUpTo (G a) := by intro H; exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ R p (G ↑a) let b' : { t // p.c t ∉ p.iUnionUpTo (G a) } := ⟨p.index (G b), B⟩ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } ⊢ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ R p (G ↑a) apply @le_ciSup _ _ _ (fun t : { t // p.c t ∉ p.iUnionUpTo (G a) } => p.r t) _ b' α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } ⊢ BddAbove (range fun t => BallPackage.r p.toBallPackage ↑t) refine' ⟨p.r_bound, fun t ht => _⟩ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ ht : t ∈ range fun t => BallPackage.r p.toBallPackage ↑t ⊢ t ≤ p.r_bound simp only [exists_prop, mem_range, Subtype.exists, Subtype.coe_mk] at ht α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ ht : ∃ a_1, ¬BallPackage.c p.toBallPackage a_1 ∈ iUnionUpTo p (if ↑a = N then i else g ↑a) ∧ BallPackage.r p.toBallPackage a_1 = t ⊢ t ≤ p.r_bound rcases ht with ⟨u, hu⟩ case intro α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ u : β hu : ¬BallPackage.c p.toBallPackage u ∈ iUnionUpTo p (if ↑a = N then i else g ↑a) ∧ BallPackage.r p.toBallPackage u = t ⊢ t ≤ p.r_bound rw [← hu.2] case intro α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N B : ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) b' : { t // ¬BallPackage.c p.toBallPackage t ∈ iUnionUpTo p (G ↑a) } := { val := index p (G ↑b), property := B } t : ℝ u : β hu : ¬BallPackage.c p.toBallPackage u ∈ iUnionUpTo p (if ↑a = N then i else g ↑a) ∧ BallPackage.r p.toBallPackage u = t ⊢ BallPackage.r p.toBallPackage u ≤ p.r_bound exact p.r_le _ α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N ⊢ ¬BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) intro H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) a b : Fin (Nat.succ N) G_lt : G ↑a < G ↑b ha : ↑a ≤ N hb : ↑b ≤ N H : BallPackage.c p.toBallPackage (index p (G ↑b)) ∈ iUnionUpTo p (G ↑a) ⊢ False exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H) α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) ⊢ ∀ (i j : Fin (Nat.succ N)), i ≠ j → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) j) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) j ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) j ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) j) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) j intro a b a_ne_b α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b wlog G_le : G a ≤ G b generalizing a b α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b G_lt : G ↑a < G ↑b ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b exact Or.inl (Gab a b G_lt) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b this : ∀ (a b : Fin (Nat.succ N)), a ≠ b → G ↑a ≤ G ↑b → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b G_le : ¬G ↑a ≤ G ↑b ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ∨ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) b) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) b exact (this b a a_ne_b.symm (le_of_not_le G_le)).symm α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b ⊢ G ↑a < G ↑b rcases G_le.lt_or_eq with (H \| H) case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a = G ↑b ⊢ G ↑a < G ↑b have A : (a : ℕ) ≠ b := Fin.val_injective.ne a_ne_b case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A✝ : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A✝) N_mem : N ∈ univ \ A✝ Inf_eq_N : sInf (univ \ A✝) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a = G ↑b A : ↑a ≠ ↑b ⊢ G ↑a < G ↑b rw [← color_G a (Nat.lt_succ_iff.1 a.2), ← color_G b (Nat.lt_succ_iff.1 b.2), H] at A case inr α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A✝ : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A✝) N_mem : N ∈ univ \ A✝ Inf_eq_N : sInf (univ \ A✝) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a = G ↑b A : color p (G ↑b) ≠ color p (G ↑b) ⊢ G ↑a < G ↑b exact (A rfl).elim case inl α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a b : Fin (Nat.succ N) a_ne_b : a ≠ b G_le : G ↑a ≤ G ↑b H : G ↑a < G ↑b ⊢ G ↑a < G ↑b exact H α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) ⊢ ∀ (i : Fin (N + 1)), i < last N → (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i intro a ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a have I : (a : ℕ) < N := ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a have : G a < G (Fin.last N) := by dsimp; simp [I.ne, (hg a I).1] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N this : G ↑a < G ↑(last N) ⊢ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a ≤ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ∧ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) ≤ p.τ * (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a exact Gab _ _ this α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ G ↑a < G ↑(last N) dsimp α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ (if ↑a = N then i else g ↑a) < if N = N then i else g N simp [I.ne, (hg a I).1] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) ⊢ ∀ (i : Fin (N + 1)), i < last N → dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) i) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) i + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) intro a ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) have I : (a : ℕ) < N := ha α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) have J : G (Fin.last N) = i := by dsimp; simp only [if_true, eq_self_iff_true] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) have K : G a = g a := by dsimp; simp [I.ne, (hg a I).1] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i K : G ↑a = g ↑a ⊢ dist ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) a) ((fun k => BallPackage.c p.toBallPackage (index p (G ↑k))) (last N)) ≤ (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) a + (fun k => BallPackage.r p.toBallPackage (index p (G ↑k))) (last N) convert dist_le_add_of_nonempty_closedBall_inter_closedBall (hg _ I).2.1 α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ G ↑(last N) = i dsimp α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N ⊢ (if N = N then i else g N) = i simp only [if_true, eq_self_iff_true] α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i ⊢ G ↑a = g ↑a dsimp α : Type u_1 inst✝¹ : MetricSpace α β : Type u inst✝ : Nonempty β p : TauPackage β α i✝ : Ordinal.{u} hi✝ : i✝ < lastStep p N : ℕ hN : IsEmpty (SatelliteConfig α N p.τ) i : Ordinal.{u} IH : ∀ (k : Ordinal.{u}), k < i → k < lastStep p → color p k < N hi : i < lastStep p A : Set ℕ := ⋃ j, ⋃ (_ : Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p ↑j)) (BallPackage.r p.toBallPackage (index p ↑j)) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))), {color p ↑j} color_i : color p i = sInf (univ \ A) N_mem : N ∈ univ \ A Inf_eq_N : sInf (univ \ A) = N g : ℕ → Ordinal.{u} hg : ∀ (k : ℕ), k < N → g k < i ∧ Set.Nonempty (closedBall (BallPackage.c p.toBallPackage (index p (g k))) (BallPackage.r p.toBallPackage (index p (g k))) ∩ closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) ∧ k = color p (g k) G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n color_G : ∀ (n : ℕ), n ≤ N → color p (G n) = n G_lt_last : ∀ (n : ℕ), n ≤ N → G n < lastStep p fGn : ∀ (n : ℕ), n ≤ N → ¬BallPackage.c p.toBallPackage (index p (G n)) ∈ iUnionUpTo p (G n) ∧ R p (G n) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G n)) Gab : ∀ (a b : Fin (Nat.succ N)), G ↑a < G ↑b → BallPackage.r p.toBallPackage (index p (G ↑a)) ≤ dist (BallPackage.c p.toBallPackage (index p (G ↑a))) (BallPackage.c p.toBallPackage (index p (G ↑b))) ∧ BallPackage.r p.toBallPackage (index p (G ↑b)) ≤ p.τ * BallPackage.r p.toBallPackage (index p (G ↑a)) a : Fin (N + 1) ha : a < last N I : ↑a < N J : G ↑(last N) = i ⊢ (if ↑a = N then i else g ↑a) = g ↑a simp [I.ne, (hg a I).1] Qed
Besicovitch.ae_tendsto_rnDeriv α : Type u_1 inst✝¹¹ : MetricSpace α β : Type u inst✝¹⁰ : SecondCountableTopology α inst✝⁹ : MeasurableSpace α inst✝⁸ : OpensMeasurableSpace α inst✝⁷ : HasBesicovitchCovering α inst✝⁶ : MetricSpace β inst✝⁵ : MeasurableSpace β inst✝⁴ : BorelSpace β inst✝³ : SecondCountableTopology β inst✝² : HasBesicovitchCovering β ρ μ : Measure β inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsLocallyFiniteMeasure ρ ⊢ ∀ᵐ (x : β) ∂μ, Tendsto (fun r => ↑↑ρ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (Measure.rnDeriv ρ μ x)) filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (Besicovitch.vitaliFamily μ) ρ] with x hx case h α : Type u_1 inst✝¹¹ : MetricSpace α β : Type u inst✝¹⁰ : SecondCountableTopology α inst✝⁹ : MeasurableSpace α inst✝⁸ : OpensMeasurableSpace α inst✝⁷ : HasBesicovitchCovering α inst✝⁶ : MetricSpace β inst✝⁵ : MeasurableSpace β inst✝⁴ : BorelSpace β inst✝³ : SecondCountableTopology β inst✝² : HasBesicovitchCovering β ρ μ : Measure β inst✝¹ : IsLocallyFiniteMeasure μ inst✝ : IsLocallyFiniteMeasure ρ x : β hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x) (𝓝 (Measure.rnDeriv ρ μ x)) ⊢ Tendsto (fun r => ↑↑ρ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (Measure.rnDeriv ρ μ x)) exact hx.comp (tendsto_filterAt μ x) ** Qed
Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β hs : MeasurableSet s ⊢ ∀ᵐ (x : β) ∂μ, Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (indicator s 1 x)) filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet (Besicovitch.vitaliFamily μ) hs] case h α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β hs : MeasurableSet s ⊢ ∀ (a : β), Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) a) (𝓝 (indicator s 1 a)) → Tendsto (fun r => ↑↑μ (s ∩ closedBall a r) / ↑↑μ (closedBall a r)) (𝓝[Ioi 0] 0) (𝓝 (indicator s 1 a)) intro x hx case h α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β hs : MeasurableSet s x : β hx : Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x) (𝓝 (indicator s 1 x)) ⊢ Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (indicator s 1 x)) exact hx.comp (tendsto_filterAt μ x) ** Qed
Besicovitch.ae_tendsto_measure_inter_div α : Type u_1 inst✝¹⁰ : MetricSpace α β : Type u inst✝⁹ : SecondCountableTopology α inst✝⁸ : MeasurableSpace α inst✝⁷ : OpensMeasurableSpace α inst✝⁶ : HasBesicovitchCovering α inst✝⁵ : MetricSpace β inst✝⁴ : MeasurableSpace β inst✝³ : BorelSpace β inst✝² : SecondCountableTopology β inst✝¹ : HasBesicovitchCovering β μ : Measure β inst✝ : IsLocallyFiniteMeasure μ s : Set β ⊢ ∀ᵐ (x : β) ∂Measure.restrict μ s, Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1) filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div (Besicovitch.vitaliFamily μ) s] with x hx using hx.comp (tendsto_filterAt μ x) ** Qed
MeasureTheory.tendstoInMeasure_iff_norm α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : SeminormedAddCommGroup E l : Filter ι f : ι → α → E g : α → E ⊢ TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ), 0 < ε → Tendsto (fun i => ↑↑μ {x \| ε ≤ ‖f i x - g x‖}) l (𝓝 0) simp_rw [TendstoInMeasure, dist_eq_norm] ** Qed
MeasureTheory.TendstoInMeasure.congr' α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ⊢ TendstoInMeasure μ f' l g' intro ε hε α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε ⊢ Tendsto (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) l (𝓝 0) suffices (fun i => μ { x \| ε ≤ dist (f' i x) (g' x) }) =ᶠ[l] fun i => μ { x \| ε ≤ dist (f i x) (g x) } by rw [tendsto_congr' this] exact h_tendsto ε hε α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε ⊢ (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) =ᶠ[l] fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)} filter_upwards [h_left] with i h_ae_eq case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i ⊢ ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)} = ↑↑μ {x \| ε ≤ dist (f i x) (g x)} refine' measure_congr _ case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i ⊢ {x \| ε ≤ dist (f' i x) (g' x)} =ᵐ[μ] {x \| ε ≤ dist (f i x) (g x)} filter_upwards [h_ae_eq, h_right] with x hxf hxg case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i x : α hxf : f i x = f' i x hxg : g x = g' x ⊢ setOf (fun x => ε ≤ dist (f' i x) (g' x)) x = setOf (fun x => ε ≤ dist (f i x) (g x)) x rw [eq_iff_iff] case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i x : α hxf : f i x = f' i x hxg : g x = g' x ⊢ setOf (fun x => ε ≤ dist (f' i x) (g' x)) x ↔ setOf (fun x => ε ≤ dist (f i x) (g x)) x change ε ≤ dist (f' i x) (g' x) ↔ ε ≤ dist (f i x) (g x) case h α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε i : ι h_ae_eq : f i =ᵐ[μ] f' i x : α hxf : f i x = f' i x hxg : g x = g' x ⊢ ε ≤ dist (f' i x) (g' x) ↔ ε ≤ dist (f i x) (g x) rw [hxg, hxf] α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε this : (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) =ᶠ[l] fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)} ⊢ Tendsto (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) l (𝓝 0) rw [tendsto_congr' this] α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : Dist E l : Filter ι f f' : ι → α → E g g' : α → E h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i h_right : g =ᵐ[μ] g' h_tendsto : TendstoInMeasure μ f l g ε : ℝ hε : 0 < ε this : (fun i => ↑↑μ {x \| ε ≤ dist (f' i x) (g' x)}) =ᶠ[l] fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)} ⊢ Tendsto (fun i => ↑↑μ {x \| ε ≤ dist (f i x) (g x)}) l (𝓝 0) exact h_tendsto ε hε ** Qed
MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ⊢ TendstoInMeasure μ f atTop g refine' fun ε hε => ENNReal.tendsto_atTop_zero.mpr fun δ hδ => _ α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0∞ hδ : δ > 0 ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ δ by_cases hδi : δ = ∞ case neg α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0∞ hδ : δ > 0 hδi : ¬δ = ⊤ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ δ lift δ to ℝ≥0 using hδi case neg.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : ↑δ > 0 ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ rw [gt_iff_lt, ENNReal.coe_pos, ← NNReal.coe_pos] at hδ case neg.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ obtain ⟨t, _, ht, hunif⟩ := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg hδ case neg.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ENNReal.ofReal ↑δ hunif : TendstoUniformlyOn (fun n => f n) g atTop tᶜ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ rw [ENNReal.ofReal_coe_nnreal] at ht case neg.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : TendstoUniformlyOn (fun n => f n) g atTop tᶜ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ rw [Metric.tendstoUniformlyOn_iff] at hunif case neg.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ obtain ⟨N, hN⟩ := eventually_atTop.1 (hunif ε hε) case neg.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ refine' ⟨N, fun n hn => _⟩ case neg.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ suffices : { x : α \| ε ≤ dist (f n x) (g x) } ⊆ t case neg.intro.intro.intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N this : {x \| ε ≤ dist (f n x) (g x)} ⊆ t ⊢ ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ ↑δ case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ {x \| ε ≤ dist (f n x) (g x)} ⊆ t exact (measure_mono this).trans ht case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ {x \| ε ≤ dist (f n x) (g x)} ⊆ t rw [← Set.compl_subset_compl] case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N ⊢ tᶜ ⊆ {x \| ε ≤ dist (f n x) (g x)}ᶜ intro x hx case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N x : α hx : x ∈ tᶜ ⊢ x ∈ {x \| ε ≤ dist (f n x) (g x)}ᶜ rw [Set.mem_compl_iff, Set.nmem_setOf_iff, dist_comm, not_le] case this α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0 hδ : 0 < ↑δ t : Set α left✝ : MeasurableSet t ht : ↑↑μ t ≤ ↑δ hunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε N : ℕ hN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε n : ℕ hn : n ≥ N x : α hx : x ∈ tᶜ ⊢ dist (g x) (f n x) < ε exact hN n hn x hx case pos α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝¹ : MetricSpace E f : ℕ → α → E g : α → E inst✝ : IsFiniteMeasure μ hf : ∀ (n : ℕ), StronglyMeasurable (f n) hg : StronglyMeasurable g hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) ε : ℝ hε : 0 < ε δ : ℝ≥0∞ hδ : δ > 0 hδi : δ = ⊤ ⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x \| ε ≤ dist (f n x) (g x)} ≤ δ simp only [hδi, imp_true_iff, le_top, exists_const] ** Qed
MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g n : ℕ ⊢ ∃ N, ∀ (m_1 : ℕ), m_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n specialize hfg ((2⁻¹ : ℝ) ^ n) (by simp only [Real.rpow_nat_cast, inv_pos, zero_lt_two, pow_pos]) α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E n : ℕ hfg : Tendsto (fun i => ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f i x) (g x)}) atTop (𝓝 0) ⊢ ∃ N, ∀ (m_1 : ℕ), m_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n rw [ENNReal.tendsto_atTop_zero] at hfg α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E n : ℕ hfg : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n_1 : ℕ), n_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f n_1 x) (g x)} ≤ ε ⊢ ∃ N, ∀ (m_1 : ℕ), m_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n exact hfg ((2 : ℝ≥0∞)⁻¹ ^ n) (pos_iff_ne_zero.mpr fun h_zero => by simpa using pow_eq_zero h_zero) α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g n : ℕ ⊢ 0 < 2⁻¹ ^ n simp only [Real.rpow_nat_cast, inv_pos, zero_lt_two, pow_pos] α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E n : ℕ hfg : ∀ (ε : ℝ≥0∞), ε > 0 → ∃ N, ∀ (n_1 : ℕ), n_1 ≥ N → ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f n_1 x) (g x)} ≤ ε h_zero : 2⁻¹ ^ n = 0 ⊢ False simpa using pow_eq_zero h_zero ** Qed
MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g n k : ℕ hn : seqTendstoAeSeq hfg n ≤ k ⊢ ↑↑μ {x \| 2⁻¹ ^ n ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ n cases n case zero α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g k : ℕ hn : seqTendstoAeSeq hfg Nat.zero ≤ k ⊢ ↑↑μ {x \| 2⁻¹ ^ Nat.zero ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ Nat.zero exact Classical.choose_spec (exists_nat_measure_lt_two_inv hfg 0) k hn case succ α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝ : MetricSpace E f : ℕ → α → E g : α → E hfg : TendstoInMeasure μ f atTop g k n✝ : ℕ hn : seqTendstoAeSeq hfg (Nat.succ n✝) ≤ k ⊢ ↑↑μ {x \| 2⁻¹ ^ Nat.succ n✝ ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ Nat.succ n✝ exact Classical.choose_spec (exists_nat_measure_lt_two_inv hfg _) _ (le_trans (le_max_left _ _) hn) ** Qed
MeasureTheory.TendstoInMeasure.exists_seq_tendstoInMeasure_atTop α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ⊢ ∃ ns, TendstoInMeasure μ (fun n => f (ns n)) atTop g obtain ⟨ns, h_tendsto_ns⟩ : ∃ ns : ℕ → ι, Tendsto ns atTop u := exists_seq_tendsto u case intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ns : ℕ → ι h_tendsto_ns : Tendsto ns atTop u ⊢ ∃ ns, TendstoInMeasure μ (fun n => f (ns n)) atTop g exact ⟨ns, fun ε hε => (hfg ε hε).comp h_tendsto_ns⟩ ** Qed
MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae' α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ⊢ ∃ ns, ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) obtain ⟨ms, hms⟩ := hfg.exists_seq_tendstoInMeasure_atTop case intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ms : ℕ → ι hms : TendstoInMeasure μ (fun n => f (ms n)) atTop g ⊢ ∃ ns, ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) obtain ⟨ns, -, hns⟩ := hms.exists_seq_tendsto_ae case intro.intro.intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝² : MetricSpace E f✝ : ℕ → α → E g✝ : α → E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hfg : TendstoInMeasure μ f u g ms : ℕ → ι hms : TendstoInMeasure μ (fun n => f (ms n)) atTop g ns : ℕ → ℕ hns : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ms (ns i)) x) atTop (𝓝 (g x)) ⊢ ∃ ns, ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) exact ⟨ms ∘ ns, hns⟩ ** Qed
MeasureTheory.TendstoInMeasure.aemeasurable α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : BorelSpace E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : TendstoInMeasure μ f u g ⊢ AEMeasurable g obtain ⟨ns, hns⟩ := h_tendsto.exists_seq_tendsto_ae' case intro α : Type u_1 ι : Type u_2 E : Type u_3 m : MeasurableSpace α μ : Measure α inst✝⁴ : MeasurableSpace E inst✝³ : NormedAddCommGroup E inst✝² : BorelSpace E u : Filter ι inst✝¹ : NeBot u inst✝ : IsCountablyGenerated u f : ι → α → E g : α → E hf : ∀ (n : ι), AEMeasurable (f n) h_tendsto : TendstoInMeasure μ f u g ns : ℕ → ι hns : ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) ⊢ AEMeasurable g exact aemeasurable_of_tendsto_metrizable_ae atTop (fun n => hf (ns n)) hns ** Qed
MeasureTheory.AEEqFun.mk_coeFn α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = f conv_rhs => rw [← Quotient.out_eq' f] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' (Quotient.out' f) set g : { f : α → β // AEStronglyMeasurable f μ } := Quotient.out' f α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : { f // AEStronglyMeasurable f μ } := Quotient.out' f ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g have : g = ⟨g.1, g.2⟩ := Subtype.eq rfl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : { f // AEStronglyMeasurable f μ } := Quotient.out' f this : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) } ⊢ mk ↑f (_ : AEStronglyMeasurable (↑f) μ) = Quotient.mk'' g rw [this, ← mk, mk_eq_mk] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : { f // AEStronglyMeasurable f μ } := Quotient.out' f this : g = { val := ↑g, property := (_ : AEStronglyMeasurable (↑g) μ) } ⊢ ↑f =ᵐ[μ] ↑g exact (AEStronglyMeasurable.ae_eq_mk _).symm ** Qed
MeasureTheory.AEEqFun.ext α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : α →ₘ[μ] β h : ↑f =ᵐ[μ] ↑g ⊢ f = g rwa [← f.mk_coeFn, ← g.mk_coeFn, mk_eq_mk] ** Qed
MeasureTheory.AEEqFun.ext_iff α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : α →ₘ[μ] β h : f = g ⊢ ↑f =ᵐ[μ] ↑g rw [h] ** Qed
MeasureTheory.AEEqFun.coeFn_mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α → β hf : AEStronglyMeasurable f μ ⊢ ↑(mk f hf) =ᵐ[μ] f apply (AEStronglyMeasurable.ae_eq_mk _).symm.trans α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α → β hf : AEStronglyMeasurable f μ ⊢ ↑(Quotient.out' (mk f hf)) =ᵐ[μ] f exact @Quotient.mk_out' _ (μ.aeEqSetoid β) (⟨f, hf⟩ : { f // AEStronglyMeasurable f μ }) ** Qed
MeasureTheory.AEEqFun.comp_eq_mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β → γ hg : Continuous g f : α →ₘ[μ] β ⊢ comp g hg f = mk (g ∘ ↑f) (_ : AEStronglyMeasurable (fun x => g (↑f x)) μ) rw [← comp_mk g hg f f.aestronglyMeasurable, mk_coeFn] ** Qed
MeasureTheory.AEEqFun.compMeasurable_eq_mk α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹⁰ : MeasurableSpace α μ ν : Measure α inst✝⁹ : TopologicalSpace β inst✝⁸ : TopologicalSpace γ inst✝⁷ : TopologicalSpace δ inst✝⁶ : MeasurableSpace β inst✝⁵ : PseudoMetrizableSpace β inst✝⁴ : BorelSpace β inst✝³ : MeasurableSpace γ inst✝² : PseudoMetrizableSpace γ inst✝¹ : OpensMeasurableSpace γ inst✝ : SecondCountableTopology γ g : β → γ hg : Measurable g f : α →ₘ[μ] β ⊢ compMeasurable g hg f = mk (g ∘ ↑f) (_ : AEStronglyMeasurable (g ∘ ↑f) μ) rw [← compMeasurable_mk g hg f f.aestronglyMeasurable, mk_coeFn] ** Qed
MeasureTheory.AEEqFun.coeFn_pair α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : α →ₘ[μ] γ ⊢ ↑(pair f g) =ᵐ[μ] fun x => (↑f x, ↑g x) rw [pair_eq_mk] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f : α →ₘ[μ] β g : α →ₘ[μ] γ ⊢ ↑(mk (fun x => (↑f x, ↑g x)) (_ : AEStronglyMeasurable (fun x => (↑f x, ↑g x)) μ)) =ᵐ[μ] fun x => (↑f x, ↑g x) apply coeFn_mk ** Qed
MeasureTheory.AEEqFun.toGerm_injective α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f g : α →ₘ[μ] β H : toGerm f = toGerm g ⊢ ↑↑f = ↑↑g rwa [← toGerm_eq, ← toGerm_eq] ** Qed
MeasureTheory.AEEqFun.comp_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β → γ hg : Continuous g f✝ : α →ₘ[μ] β f : α → β x✝ : AEStronglyMeasurable f μ ⊢ toGerm (comp g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝)) simp ** Qed
MeasureTheory.AEEqFun.compMeasurable_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹⁰ : MeasurableSpace α μ ν : Measure α inst✝⁹ : TopologicalSpace β inst✝⁸ : TopologicalSpace γ inst✝⁷ : TopologicalSpace δ inst✝⁶ : MeasurableSpace β inst✝⁵ : BorelSpace β inst✝⁴ : PseudoMetrizableSpace β inst✝³ : PseudoMetrizableSpace γ inst✝² : SecondCountableTopology γ inst✝¹ : MeasurableSpace γ inst✝ : OpensMeasurableSpace γ g : β → γ hg : Measurable g f✝ : α →ₘ[μ] β f : α → β x✝ : AEStronglyMeasurable f μ ⊢ toGerm (compMeasurable g hg (mk f x✝)) = Germ.map g (toGerm (mk f x✝)) simp ** Qed
MeasureTheory.AEEqFun.comp₂_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ g : β → γ → δ hg : Continuous (uncurry g) f₁✝ : α →ₘ[μ] β f₂✝ : α →ₘ[μ] γ f₁ : α → β x✝¹ : AEStronglyMeasurable f₁ μ f₂ : α → γ x✝ : AEStronglyMeasurable f₂ μ ⊢ toGerm (comp₂ g hg (mk f₁ x✝¹) (mk f₂ x✝)) = Germ.map₂ g (toGerm (mk f₁ x✝¹)) (toGerm (mk f₂ x✝)) simp ** Qed
MeasureTheory.AEEqFun.comp₂Measurable_toGerm α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝¹⁴ : MeasurableSpace α μ ν : Measure α inst✝¹³ : TopologicalSpace β inst✝¹² : TopologicalSpace γ inst✝¹¹ : TopologicalSpace δ inst✝¹⁰ : PseudoMetrizableSpace β inst✝⁹ : MeasurableSpace β inst✝⁸ : BorelSpace β inst✝⁷ : PseudoMetrizableSpace γ inst✝⁶ : SecondCountableTopologyEither β γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : PseudoMetrizableSpace δ inst✝² : SecondCountableTopology δ inst✝¹ : MeasurableSpace δ inst✝ : OpensMeasurableSpace δ g : β → γ → δ hg : Measurable (uncurry g) f₁✝ : α →ₘ[μ] β f₂✝ : α →ₘ[μ] γ f₁ : α → β x✝¹ : AEStronglyMeasurable f₁ μ f₂ : α → γ x✝ : AEStronglyMeasurable f₂ μ ⊢ toGerm (comp₂Measurable g hg (mk f₁ x✝¹) (mk f₂ x✝)) = Germ.map₂ g (toGerm (mk f₁ x✝¹)) (toGerm (mk f₂ x✝)) simp ** Qed
MeasureTheory.AEEqFun.le_sup_left α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ f ≤ f ⊔ g rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ ↑f ≤ᵐ[μ] ↑(f ⊔ g) filter_upwards [coeFn_sup f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑f a✝ ≤ ↑(f ⊔ g) a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑f a✝ ≤ ↑f a✝ ⊔ ↑g a✝ exact le_sup_left ** Qed
MeasureTheory.AEEqFun.le_sup_right α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ g ≤ f ⊔ g rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β ⊢ ↑g ≤ᵐ[μ] ↑(f ⊔ g) filter_upwards [coeFn_sup f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑g a✝ ≤ ↑(f ⊔ g) a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑g a✝ ≤ ↑f a✝ ⊔ ↑g a✝ exact le_sup_right ** Qed
MeasureTheory.AEEqFun.sup_le α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : f ≤ f' hg : g ≤ f' ⊢ f ⊔ g ≤ f' rw [← coeFn_le] at hf hg ⊢ α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : ↑f ≤ᵐ[μ] ↑f' hg : ↑g ≤ᵐ[μ] ↑f' ⊢ ↑(f ⊔ g) ≤ᵐ[μ] ↑f' filter_upwards [hf, hg, coeFn_sup f g] with _ haf hag ha_sup case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : ↑f ≤ᵐ[μ] ↑f' hg : ↑g ≤ᵐ[μ] ↑f' a✝ : α haf : ↑f a✝ ≤ ↑f' a✝ hag : ↑g a✝ ≤ ↑f' a✝ ha_sup : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑(f ⊔ g) a✝ ≤ ↑f' a✝ rw [ha_sup] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeSup β inst✝ : ContinuousSup β f g f' : α →ₘ[μ] β hf : ↑f ≤ᵐ[μ] ↑f' hg : ↑g ≤ᵐ[μ] ↑f' a✝ : α haf : ↑f a✝ ≤ ↑f' a✝ hag : ↑g a✝ ≤ ↑f' a✝ ha_sup : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ ⊢ ↑f a✝ ⊔ ↑g a✝ ≤ ↑f' a✝ exact sup_le haf hag ** Qed
MeasureTheory.AEEqFun.inf_le_left α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ f ⊓ g ≤ f rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑f filter_upwards [coeFn_inf f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝ exact inf_le_left ** Qed
MeasureTheory.AEEqFun.inf_le_right α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ f ⊓ g ≤ g rw [← coeFn_le] α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β ⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑g filter_upwards [coeFn_inf f g] with _ ha case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝ rw [ha] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f g : α →ₘ[μ] β a✝ : α ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑g a✝ exact inf_le_right ** Qed
MeasureTheory.AEEqFun.le_inf α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : f' ≤ f hg : f' ≤ g ⊢ f' ≤ f ⊓ g rw [← coeFn_le] at hf hg ⊢ α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : ↑f' ≤ᵐ[μ] ↑f hg : ↑f' ≤ᵐ[μ] ↑g ⊢ ↑f' ≤ᵐ[μ] ↑(f ⊓ g) filter_upwards [hf, hg, coeFn_inf f g] with _ haf hag ha_inf case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : ↑f' ≤ᵐ[μ] ↑f hg : ↑f' ≤ᵐ[μ] ↑g a✝ : α haf : ↑f' a✝ ≤ ↑f a✝ hag : ↑f' a✝ ≤ ↑g a✝ ha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f' a✝ ≤ ↑(f ⊓ g) a✝ rw [ha_inf] case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝⁵ : MeasurableSpace α μ ν : Measure α inst✝⁴ : TopologicalSpace β inst✝³ : TopologicalSpace γ inst✝² : TopologicalSpace δ inst✝¹ : SemilatticeInf β inst✝ : ContinuousInf β f' f g : α →ₘ[μ] β hf : ↑f' ≤ᵐ[μ] ↑f hg : ↑f' ≤ᵐ[μ] ↑g a✝ : α haf : ↑f' a✝ ≤ ↑f a✝ hag : ↑f' a✝ ≤ ↑g a✝ ha_inf : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ ⊢ ↑f' a✝ ≤ ↑f a✝ ⊓ ↑g a✝ exact le_inf haf hag ** Qed
MeasureTheory.AEEqFun.lintegral_add α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 inst✝³ : MeasurableSpace α μ ν : Measure α inst✝² : TopologicalSpace β inst✝¹ : TopologicalSpace γ inst✝ : TopologicalSpace δ f✝ g✝ : α →ₘ[μ] ℝ≥0∞ f : α → ℝ≥0∞ hf : AEStronglyMeasurable f μ g : α → ℝ≥0∞ x✝ : AEStronglyMeasurable g μ ⊢ lintegral (mk f hf + mk g x✝) = lintegral (mk f hf) + lintegral (mk g x✝) simp [lintegral_add_left' hf.aemeasurable] ** Qed
MeasureTheory.Measure.haveLebesgueDecomposition_spec α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ Measurable (rnDeriv μ ν) ∧ singularPart μ ν ⟂ₘ ν ∧ μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) rw [singularPart, rnDeriv, dif_pos h, dif_pos h] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ Measurable (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).2 ∧ (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).1 ⟂ₘ ν ∧ μ = (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).1 + withDensity ν (Classical.choose (_ : ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + withDensity ν p.2)).2 exact Classical.choose_spec h.lebesgue_decomposition ** Qed
MeasureTheory.Measure.measurable_rnDeriv α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ Measurable (rnDeriv μ ν) by_cases h : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ Measurable (rnDeriv μ ν) exact (haveLebesgueDecomposition_spec μ ν).1 case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ Measurable (rnDeriv μ ν) rw [rnDeriv, dif_neg h] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ Measurable 0 exact measurable_zero ** Qed
MeasureTheory.Measure.mutuallySingular_singularPart α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ singularPart μ ν ⟂ₘ ν by_cases h : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ⟂ₘ ν exact (haveLebesgueDecomposition_spec μ ν).2.1 case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ⟂ₘ ν rw [singularPart, dif_neg h] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α h : ¬HaveLebesgueDecomposition μ ν ⊢ 0 ⟂ₘ ν exact MutuallySingular.zero_left ** Qed
MeasureTheory.Measure.singularPart_le α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ singularPart μ ν ≤ μ by_cases hl : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ≤ μ cases' (haveLebesgueDecomposition_spec μ ν).2 with _ h case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ singularPart μ ν ≤ μ conv_rhs => rw [h] case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ singularPart μ ν ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν) exact Measure.le_add_right le_rfl case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ singularPart μ ν ≤ μ rw [singularPart, dif_neg hl] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ 0 ≤ μ exact Measure.zero_le μ ** Qed
MeasureTheory.Measure.withDensity_rnDeriv_le α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α ⊢ withDensity ν (rnDeriv μ ν) ≤ μ by_cases hl : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν ⊢ withDensity ν (rnDeriv μ ν) ≤ μ cases' (haveLebesgueDecomposition_spec μ ν).2 with _ h case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ withDensity ν (rnDeriv μ ν) ≤ μ conv_rhs => rw [h] case pos.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : HaveLebesgueDecomposition μ ν left✝ : singularPart μ ν ⟂ₘ ν h : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ withDensity ν (rnDeriv μ ν) ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν) exact Measure.le_add_left le_rfl case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ withDensity ν (rnDeriv μ ν) ≤ μ rw [rnDeriv, dif_neg hl, withDensity_zero] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α hl : ¬HaveLebesgueDecomposition μ ν ⊢ 0 ≤ μ exact Measure.zero_le μ ** Qed
MeasureTheory.Measure.lintegral_rnDeriv_lt_top_of_measure_ne_top α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ by_cases hl : HaveLebesgueDecomposition μ ν case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl : HaveLebesgueDecomposition μ ν ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ haveI := hl case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ obtain ⟨-, -, hadd⟩ := haveLebesgueDecomposition_spec μ ν case pos.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ suffices : (∫⁻ x in toMeasurable μ s, μ.rnDeriv ν x ∂ν) < ∞ case pos.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this✝ : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) this : ∫⁻ (x : α) in toMeasurable μ s, rnDeriv μ ν x ∂ν < ⊤ ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ∫⁻ (x : α) in toMeasurable μ s, rnDeriv μ ν x ∂ν < ⊤ exact lt_of_le_of_lt (lintegral_mono_set (subset_toMeasurable _ _)) this case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ∫⁻ (x : α) in toMeasurable μ s, rnDeriv μ ν x ∂ν < ⊤ rw [← withDensity_apply _ (measurableSet_toMeasurable _ _)] case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ↑↑(withDensity ν fun x => rnDeriv μ ν x) (toMeasurable μ s) < ⊤ refine' lt_of_le_of_lt (le_add_left le_rfl : _ ≤ μ.singularPart ν (toMeasurable μ s) + ν.withDensity (μ.rnDeriv ν) (toMeasurable μ s)) _ case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ↑↑(singularPart μ ν) (toMeasurable μ s) + ↑↑(withDensity ν (rnDeriv μ ν)) (toMeasurable μ s) < ⊤ rw [← Measure.add_apply, ← hadd, measure_toMeasurable] case this α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl this : HaveLebesgueDecomposition μ ν hadd : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ ↑↑μ s < ⊤ exact hs.lt_top case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl : ¬HaveLebesgueDecomposition μ ν ⊢ ∫⁻ (x : α) in s, rnDeriv μ ν x ∂ν < ⊤ erw [Measure.rnDeriv, dif_neg hl, lintegral_zero] case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ hl : ¬HaveLebesgueDecomposition μ ν ⊢ 0 < ⊤ exact WithTop.zero_lt_top ** Qed
MeasureTheory.Measure.lintegral_rnDeriv_lt_top α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : IsFiniteMeasure μ ⊢ ∫⁻ (x : α), rnDeriv μ ν x ∂ν < ⊤ rw [← set_lintegral_univ] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : IsFiniteMeasure μ ⊢ ∫⁻ (x : α) in univ, rnDeriv μ ν x ∂ν < ⊤ exact lintegral_rnDeriv_lt_top_of_measure_ne_top _ (measure_lt_top _ _).ne ** Qed
MeasureTheory.Measure.rnDeriv_lt_top α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ ⊢ ∀ᵐ (x : α) ∂ν, rnDeriv μ ν x < ⊤ suffices ∀ n, ∀ᵐ x ∂ν, x ∈ spanningSets μ n → μ.rnDeriv ν x < ∞ by filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ ⊢ ∀ (n : ℕ), ∀ᵐ (x : α) ∂ν, x ∈ spanningSets μ n → rnDeriv μ ν x < ⊤ intro n α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ∀ᵐ (x : α) ∂ν, x ∈ spanningSets μ n → rnDeriv μ ν x < ⊤ rw [← ae_restrict_iff' (measurable_spanningSets _ _)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ∀ᵐ (x : α) ∂restrict ν (spanningSets μ n), rnDeriv μ ν x < ⊤ apply ae_lt_top (measurable_rnDeriv _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ∫⁻ (x : α) in spanningSets μ n, rnDeriv μ ν x ∂ν ≠ ⊤ refine' (lintegral_rnDeriv_lt_top_of_measure_ne_top _ _).ne α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ n : ℕ ⊢ ↑↑μ (spanningSets μ n) ≠ ⊤ exact (measure_spanningSets_lt_top _ _).ne α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν✝ μ ν : Measure α inst✝ : SigmaFinite μ this : ∀ (n : ℕ), ∀ᵐ (x : α) ∂ν, x ∈ spanningSets μ n → rnDeriv μ ν x < ⊤ ⊢ ∀ᵐ (x : α) ∂ν, rnDeriv μ ν x < ⊤ filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _) ** Qed
MeasureTheory.Measure.eq_singularPart α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f ⊢ s = singularPart μ ν haveI : HaveLebesgueDecomposition μ ν := ⟨⟨⟨s, f⟩, hf, hs, hadd⟩⟩ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν ⊢ s = singularPart μ ν obtain ⟨hmeas, hsing, hadd'⟩ := haveLebesgueDecomposition_spec μ ν case intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hsing : singularPart μ ν ⟂ₘ ν hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ s = singularPart μ ν obtain ⟨⟨S, hS₁, hS₂, hS₃⟩, ⟨T, hT₁, hT₂, hT₃⟩⟩ := hs, hsing case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ s = singularPart μ ν rw [hadd'] at hadd case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ s = singularPart μ ν have hνinter : ν (S ∩ T)ᶜ = 0 := by rw [compl_inter] refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) rw [hT₃, hS₃, add_zero] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ s = singularPart μ ν have heq : s.restrict (S ∩ T)ᶜ = (μ.singularPart ν).restrict (S ∩ T)ᶜ := by ext1 A hA have hf : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) have hrn : ν.withDensity (μ.rnDeriv ν) (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) rw [restrict_apply hA, restrict_apply hA, ← add_zero (s (A ∩ (S ∩ T)ᶜ)), ← hf, ← add_apply, ← hadd, add_apply, hrn, add_zero] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ ⊢ s = singularPart μ ν have heq' : ∀ A : Set α, MeasurableSet A → s A = s.restrict (S ∩ T)ᶜ A := by intro A hA have hsinter : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) rw [restrict_apply hA, ← diff_eq, AEDisjoint.measure_diff_left hsinter] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A ⊢ s = singularPart μ ν ext1 A hA case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A ⊢ ↑↑s A = ↑↑(singularPart μ ν) A have hμinter : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A hμinter : ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 ⊢ ↑↑s A = ↑↑(singularPart μ ν) A rw [heq' A hA, heq, restrict_apply hA, ← diff_eq, AEDisjoint.measure_diff_left hμinter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (S ∩ T)ᶜ = 0 rw [compl_inter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (Sᶜ ∪ Tᶜ) = 0 refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν Sᶜ + ↑↑ν Tᶜ ≤ 0 rw [hT₃, hS₃, add_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ ext1 A hA case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑(restrict s (S ∩ T)ᶜ) A = ↑↑(restrict (singularPart μ ν) (S ∩ T)ᶜ) A have hf : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(restrict s (S ∩ T)ᶜ) A = ↑↑(restrict (singularPart μ ν) (S ∩ T)ᶜ) A have hrn : ν.withDensity (μ.rnDeriv ν) (A ∩ (S ∩ T)ᶜ) = 0 := by refine' withDensity_absolutelyContinuous ν _ _ rw [← nonpos_iff_eq_zero] exact hνinter ▸ measure_mono (inter_subset_right _ _) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 hrn : ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(restrict s (S ∩ T)ᶜ) A = ↑↑(restrict (singularPart μ ν) (S ∩ T)ᶜ) A rw [restrict_apply hA, restrict_apply hA, ← add_zero (s (A ∩ (S ∩ T)ᶜ)), ← hf, ← add_apply, ← hadd, add_apply, hrn, add_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 refine' withDensity_absolutelyContinuous ν _ _ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) ≤ 0 exact hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 refine' withDensity_absolutelyContinuous ν _ _ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf✝ : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hf : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑ν (A ∩ (S ∩ T)ᶜ) ≤ 0 exact hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ ⊢ ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A intro A hA α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A ⊢ ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A have hsinter : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A hsinter : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A rw [restrict_apply hA, ← diff_eq, AEDisjoint.measure_diff_left hsinter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) ≤ 0 exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict s (S ∩ T)ᶜ = restrict (singularPart μ ν) (S ∩ T)ᶜ heq' : ∀ (A : Set α), MeasurableSet A → ↑↑s A = ↑↑(restrict s (S ∩ T)ᶜ) A A : Set α hA : MeasurableSet A ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) ≤ 0 exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) ** Qed
MeasureTheory.Measure.singularPart_zero α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ ν : Measure α ⊢ singularPart 0 ν = 0 refine' (eq_singularPart measurable_zero MutuallySingular.zero_left _).symm α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ ν : Measure α ⊢ 0 = 0 + withDensity ν 0 rw [zero_add, withDensity_zero] ** Qed
MeasureTheory.Measure.singularPart_add α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ singularPart (μ₁ + μ₂) ν = singularPart μ₁ ν + singularPart μ₂ ν refine' (eq_singularPart ((measurable_rnDeriv μ₁ ν).add (measurable_rnDeriv μ₂ ν)) ((haveLebesgueDecomposition_spec _ _).2.1.add_left (haveLebesgueDecomposition_spec _ _).2.1) _).symm α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ μ₁ + μ₂ = singularPart μ₁ ν + singularPart μ₂ ν + withDensity ν fun a => rnDeriv μ₁ ν a + rnDeriv μ₂ ν a erw [withDensity_add_left (measurable_rnDeriv μ₁ ν)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ μ₁ + μ₂ = singularPart μ₁ ν + singularPart μ₂ ν + (withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a) conv_rhs => rw [add_assoc, add_comm (μ₂.singularPart ν), ← add_assoc, ← add_assoc] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ μ₁ μ₂ ν : Measure α inst✝¹ : HaveLebesgueDecomposition μ₁ ν inst✝ : HaveLebesgueDecomposition μ₂ ν ⊢ μ₁ + μ₂ = (singularPart μ₁ ν + withDensity ν (rnDeriv μ₁ ν) + withDensity ν fun a => rnDeriv μ₂ ν a) + singularPart μ₂ ν rw [← haveLebesgueDecomposition_add μ₁ ν, add_assoc, add_comm (ν.withDensity (μ₂.rnDeriv ν)), ← haveLebesgueDecomposition_add μ₂ ν] ** Qed
MeasureTheory.Measure.singularPart_withDensity α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν✝ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f ⊢ withDensity ν f = 0 + withDensity ν f rw [zero_add] ** Qed
MeasureTheory.Measure.eq_withDensity_rnDeriv α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) haveI : HaveLebesgueDecomposition μ ν := ⟨⟨⟨s, f⟩, hf, hs, hadd⟩⟩ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) obtain ⟨hmeas, hsing, hadd'⟩ := haveLebesgueDecomposition_spec μ ν case intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hs : s ⟂ₘ ν hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hsing : singularPart μ ν ⟂ₘ ν hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) obtain ⟨⟨S, hS₁, hS₂, hS₃⟩, ⟨T, hT₁, hT₂, hT₃⟩⟩ := hs, hsing case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : μ = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) rw [hadd'] at hadd case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) have hνinter : ν (S ∩ T)ᶜ = 0 := by rw [compl_inter] refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) rw [hT₃, hS₃, add_zero] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) have heq : (ν.withDensity f).restrict (S ∩ T) = (ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) := by ext1 A hA have hs : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) have hsing : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ← add_apply, add_comm, ← hadd, add_apply, hsing, zero_add] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) have heq' : ∀ A : Set α, MeasurableSet A → ν.withDensity f A = (ν.withDensity f).restrict (S ∩ T) A := by intro A hA have hνfinter : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by rw [← nonpos_iff_eq_zero] exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono (inter_subset_right _ _) rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq, measure_inter_add_diff _ (hS₁.inter hT₁)] case intro.intro.intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A ⊢ withDensity ν f = withDensity ν (rnDeriv μ ν) ext1 A hA case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) A = ↑↑(withDensity ν (rnDeriv μ ν)) A have hνrn : ν.withDensity (μ.rnDeriv ν) (A ∩ (S ∩ T)ᶜ) = 0 := by rw [← nonpos_iff_eq_zero] exact withDensity_absolutelyContinuous ν (μ.rnDeriv ν) hνinter ▸ measure_mono (inter_subset_right _ _) case intro.intro.intro.intro.intro.intro.intro.intro.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A hνrn : ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(withDensity ν f) A = ↑↑(withDensity ν (rnDeriv μ ν)) A rw [heq' A hA, heq, ← add_zero ((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) A), ← hνrn, restrict_apply hA, ← diff_eq, measure_inter_add_diff _ (hS₁.inter hT₁)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (S ∩ T)ᶜ = 0 rw [compl_inter] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν (Sᶜ ∪ Tᶜ) = 0 refine' nonpos_iff_eq_zero.1 (le_trans (measure_union_le _ _) _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 ⊢ ↑↑ν Sᶜ + ↑↑ν Tᶜ ≤ 0 rw [hT₃, hS₃, add_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 ⊢ restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) ext1 A hA case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑(restrict (withDensity ν f) (S ∩ T)) A = ↑↑(restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T)) A have hs : s (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(restrict (withDensity ν f) (S ∩ T)) A = ↑↑(restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T)) A have hsing : μ.singularPart ν (A ∩ (S ∩ T)) = 0 := by rw [← nonpos_iff_eq_zero] exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 hsing : ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(restrict (withDensity ν f) (S ∩ T)) A = ↑↑(restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T)) A rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ← add_apply, add_comm, ← hadd, add_apply, hsing, zero_add] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A ⊢ ↑↑s (A ∩ (S ∩ T)) ≤ 0 exact hS₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_left _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 A : Set α hA : MeasurableSet A hs : ↑↑s (A ∩ (S ∩ T)) = 0 ⊢ ↑↑(singularPart μ ν) (A ∩ (S ∩ T)) ≤ 0 exact hT₂ ▸ measure_mono ((inter_subset_right _ _).trans (inter_subset_right _ _)) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) ⊢ ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A intro A hA α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A have hνfinter : ν.withDensity f (A ∩ (S ∩ T)ᶜ) = 0 := by rw [← nonpos_iff_eq_zero] exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A hνfinter : ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 ⊢ ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A rw [restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hνfinter, ← diff_eq, measure_inter_add_diff _ (hS₁.inter hT₁)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν f) (A ∩ (S ∩ T)ᶜ) ≤ 0 exact withDensity_absolutelyContinuous ν f hνinter ▸ measure_mono (inter_subset_right _ _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) = 0 rw [← nonpos_iff_eq_zero] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν s : Measure α f : α → ℝ≥0∞ hf : Measurable f hadd : singularPart μ ν + withDensity ν (rnDeriv μ ν) = s + withDensity ν f this : HaveLebesgueDecomposition μ ν hmeas : Measurable (rnDeriv μ ν) hadd' : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν) S : Set α hS₁ : MeasurableSet S hS₂ : ↑↑s S = 0 hS₃ : ↑↑ν Sᶜ = 0 T : Set α hT₁ : MeasurableSet T hT₂ : ↑↑(singularPart μ ν) T = 0 hT₃ : ↑↑ν Tᶜ = 0 hνinter : ↑↑ν (S ∩ T)ᶜ = 0 heq : restrict (withDensity ν f) (S ∩ T) = restrict (withDensity ν (rnDeriv μ ν)) (S ∩ T) heq' : ∀ (A : Set α), MeasurableSet A → ↑↑(withDensity ν f) A = ↑↑(restrict (withDensity ν f) (S ∩ T)) A A : Set α hA : MeasurableSet A ⊢ ↑↑(withDensity ν (rnDeriv μ ν)) (A ∩ (S ∩ T)ᶜ) ≤ 0 exact withDensity_absolutelyContinuous ν (μ.rnDeriv ν) hνinter ▸ measure_mono (inter_subset_right _ _) ** Qed
MeasureTheory.Measure.LebesgueDecomposition.zero_mem_measurableLE α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α A : Set α x✝ : MeasurableSet A ⊢ ∫⁻ (x : α) in A, OfNat.ofNat 0 x ∂μ ≤ ↑↑ν A simp ** Qed
MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a refine Option.ext fun x => ?_ α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 ⊢ x ∈ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ↔ x ∈ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a simp only [Option.mem_def, ENNReal.some_eq_coe] α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = ↑x ↔ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x constructor <;> intro h <;> rw [← h] case mp α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = ↑x ⊢ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a symm case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a set c := ⨆ (k : ℕ) (_ : k ≤ m + 1), f k a with hc case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 h : f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a = ↑x c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ⊢ c = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a set d := f m.succ a ⊔ ⨆ (k : ℕ) (_ : k ≤ m), f k a with hd case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ c = d rw [@le_antisymm_iff ℝ≥0∞, hc, hd] case mpr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ∧ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a refine' ⟨_, _⟩ case mpr.refine'_1 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a refine' iSup₂_le fun n hn => _ case mpr.refine'_1 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a n : ℕ hn : n ≤ m + 1 ⊢ f n a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a rcases Nat.of_le_succ hn with (h \| h) case mpr.refine'_1.inl α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h✝ : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a n : ℕ hn : n ≤ m + 1 h : n ≤ m ⊢ f n a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a exact le_sup_of_le_right (le_iSup₂ (f := fun k (_ : k ≤ m) => f k a) n h) case mpr.refine'_1.inr α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h✝ : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a n : ℕ hn : n ≤ m + 1 h : n = Nat.succ m ⊢ f n a ≤ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a exact h ▸ le_sup_left case mpr.refine'_2 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a refine' sup_le _ (biSup_mono fun n hn => hn.trans m.le_succ) case mpr.refine'_2 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ f (Nat.succ m) a ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a convert @le_iSup₂ ℝ≥0∞ ℕ (fun i => i ≤ m + 1) _ _ m.succ le_rfl case h.e'_3 α✝ : Type u_1 β : Type u_2 m✝ : MeasurableSpace α✝ μ ν : Measure α✝ α : Sort u_3 f : ℕ → α → ℝ≥0∞ m : ℕ a : α x : ℝ≥0 c : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a hc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a d : ℝ≥0∞ := f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a h : d = ↑x hd : d = f (Nat.succ m) a ⊔ ⨆ k, ⨆ (_ : k ≤ m), f k a ⊢ f (Nat.succ m) a = f (Nat.succ m) a rfl ** Qed
MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE' α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν n : ℕ ⊢ ⨆ k, ⨆ (_ : k ≤ n), f k ∈ measurableLE μ ν convert iSup_mem_measurableLE f hf n case h.e'_4.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν n : ℕ x✝ : α ⊢ iSup (fun k => ⨆ (_ : k ≤ n), f k) x✝ = ⨆ k, ⨆ (_ : k ≤ n), f k x✝ refine Option.ext fun x => ?_ case h.e'_4.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), f n ∈ measurableLE μ ν n : ℕ x✝ : α x : ℝ≥0 ⊢ x ∈ iSup (fun k => ⨆ (_ : k ≤ n), f k) x✝ ↔ x ∈ ⨆ k, ⨆ (_ : k ≤ n), f k x✝ simp ** Qed
MeasureTheory.SignedMeasure.measurable_rnDeriv α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Measurable (rnDeriv s μ) rw [rnDeriv] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).posPart μ x) - ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) measurability ** Qed
MeasureTheory.SignedMeasure.integrable_rnDeriv case refine'_2 α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Integrable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) constructor case refine'_2.right α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ HasFiniteIntegral fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) exact hasFiniteIntegral_toReal_of_lintegral_ne_top (lintegral_rnDeriv_lt_top _ μ).ne case refine'_2.left α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ AEStronglyMeasurable (fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x)) μ apply Measurable.aestronglyMeasurable case refine'_2.left.hf α : Type u_1 β : Type u_2 m : MeasurableSpace α μ✝ ν : Measure α s✝ t s : SignedMeasure α μ : Measure α ⊢ Measurable fun x => ENNReal.toReal (Measure.rnDeriv (toJordanDecomposition s).negPart μ x) measurability ** Qed