formal
stringlengths 41
427k
| informal
stringclasses 1
value |
|---|---|
MeasureTheory.withDensityᵥ_neg ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E ⊢ withDensityᵥ μ (-f) = -withDensityᵥ μ f ** by_cases hf : Integrable f μ ** case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f ⊢ withDensityᵥ μ (-f) = -withDensityᵥ μ f ** ext1 i hi ** case pos.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f i : Set α hi : MeasurableSet i ⊢ ↑(withDensityᵥ μ (-f)) i = ↑(-withDensityᵥ μ f) i ** rw [VectorMeasure.neg_apply, withDensityᵥ_apply hf hi, ← integral_neg,
withDensityᵥ_apply hf.neg hi] ** case pos.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f i : Set α hi : MeasurableSet i ⊢ ∫ (x : α) in i, (-f) x ∂μ = ∫ (a : α) in i, -f a ∂μ ** rfl ** case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : ¬Integrable f ⊢ withDensityᵥ μ (-f) = -withDensityᵥ μ f ** rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg, neg_zero] ** case neg.hnc α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : ¬Integrable f ⊢ ¬Integrable (-f) ** rwa [integrable_neg_iff] ** Qed
| |
MeasureTheory.withDensityᵥ_add ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f hg : Integrable g ⊢ withDensityᵥ μ (f + g) = withDensityᵥ μ f + withDensityᵥ μ g ** ext1 i hi ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f hg : Integrable g i : Set α hi : MeasurableSet i ⊢ ↑(withDensityᵥ μ (f + g)) i = ↑(withDensityᵥ μ f + withDensityᵥ μ g) i ** rw [withDensityᵥ_apply (hf.add hg) hi, VectorMeasure.add_apply, withDensityᵥ_apply hf hi,
withDensityᵥ_apply hg hi] ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f hg : Integrable g i : Set α hi : MeasurableSet i ⊢ ∫ (x : α) in i, (f + g) x ∂μ = ∫ (x : α) in i, f x ∂μ + ∫ (x : α) in i, g x ∂μ ** simp_rw [Pi.add_apply] ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f hg : Integrable g i : Set α hi : MeasurableSet i ⊢ ∫ (x : α) in i, f x + g x ∂μ = ∫ (x : α) in i, f x ∂μ + ∫ (x : α) in i, g x ∂μ ** rw [integral_add] <;> rw [← integrableOn_univ] ** case h.hf α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f hg : Integrable g i : Set α hi : MeasurableSet i ⊢ IntegrableOn (fun x => f x) Set.univ ** exact hf.integrableOn.restrict MeasurableSet.univ ** case h.hg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f hg : Integrable g i : Set α hi : MeasurableSet i ⊢ IntegrableOn (fun x => g x) Set.univ ** exact hg.integrableOn.restrict MeasurableSet.univ ** Qed
| |
MeasureTheory.withDensityᵥ_sub ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f hg : Integrable g ⊢ withDensityᵥ μ (f - g) = withDensityᵥ μ f - withDensityᵥ μ g ** rw [sub_eq_add_neg, sub_eq_add_neg, withDensityᵥ_add hf hg.neg, withDensityᵥ_neg] ** Qed
| |
MeasureTheory.withDensityᵥ_smul ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 ⊢ withDensityᵥ μ (r • f) = r • withDensityᵥ μ f ** by_cases hf : Integrable f μ ** case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 hf : Integrable f ⊢ withDensityᵥ μ (r • f) = r • withDensityᵥ μ f ** ext1 i hi ** case pos.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 hf : Integrable f i : Set α hi : MeasurableSet i ⊢ ↑(withDensityᵥ μ (r • f)) i = ↑(r • withDensityᵥ μ f) i ** rw [withDensityᵥ_apply (hf.smul r) hi, VectorMeasure.smul_apply, withDensityᵥ_apply hf hi, ←
integral_smul r f] ** case pos.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 hf : Integrable f i : Set α hi : MeasurableSet i ⊢ ∫ (x : α) in i, (r • f) x ∂μ = ∫ (a : α) in i, r • f a ∂μ ** rfl ** case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 hf : ¬Integrable f ⊢ withDensityᵥ μ (r • f) = r • withDensityᵥ μ f ** by_cases hr : r = 0 ** case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 hf : ¬Integrable f hr : r = 0 ⊢ withDensityᵥ μ (r • f) = r • withDensityᵥ μ f ** rw [hr, zero_smul, zero_smul, withDensityᵥ_zero] ** case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 hf : ¬Integrable f hr : ¬r = 0 ⊢ withDensityᵥ μ (r • f) = r • withDensityᵥ μ f ** rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg, smul_zero] ** case neg.hnc α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E f✝ g : α → E 𝕜 : Type u_4 inst✝² : NontriviallyNormedField 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass ℝ 𝕜 E f : α → E r : 𝕜 hf : ¬Integrable f hr : ¬r = 0 ⊢ ¬Integrable (r • f) ** rwa [integrable_smul_iff hr f] ** Qed
| |
MeasureTheory.Integrable.ae_eq_of_withDensityᵥ_eq ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E hf : Integrable f hg : Integrable g hfg : withDensityᵥ μ f = withDensityᵥ μ g ⊢ f =ᶠ[ae μ] g ** refine' hf.ae_eq_of_forall_set_integral_eq f g hg fun i hi _ => _ ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E hf : Integrable f hg : Integrable g hfg : withDensityᵥ μ f = withDensityᵥ μ g i : Set α hi : MeasurableSet i x✝ : ↑↑μ i < ⊤ ⊢ ∫ (x : α) in i, f x ∂μ = ∫ (x : α) in i, g x ∂μ ** rw [← withDensityᵥ_apply hf hi, hfg, withDensityᵥ_apply hg hi] ** Qed
| |
MeasureTheory.WithDensityᵥEq.congr_ae ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g ⊢ withDensityᵥ μ f = withDensityᵥ μ g ** by_cases hf : Integrable f μ ** case pos α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g hf : Integrable f ⊢ withDensityᵥ μ f = withDensityᵥ μ g ** ext i hi ** case pos.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g hf : Integrable f i : Set α hi : MeasurableSet i ⊢ ↑(withDensityᵥ μ f) i = ↑(withDensityᵥ μ g) i ** rw [withDensityᵥ_apply hf hi, withDensityᵥ_apply (hf.congr h) hi] ** case pos.h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g hf : Integrable f i : Set α hi : MeasurableSet i ⊢ ∫ (x : α) in i, f x ∂μ = ∫ (x : α) in i, g x ∂μ ** exact integral_congr_ae (ae_restrict_of_ae h) ** case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g hf : ¬Integrable f ⊢ withDensityᵥ μ f = withDensityᵥ μ g ** have hg : ¬Integrable g μ := by intro hg; exact hf (hg.congr h.symm) ** case neg α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g hf : ¬Integrable f hg : ¬Integrable g ⊢ withDensityᵥ μ f = withDensityᵥ μ g ** rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg hg] ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g hf : ¬Integrable f ⊢ ¬Integrable g ** intro hg ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g✝ f g : α → E h : f =ᶠ[ae μ] g hf : ¬Integrable f hg : Integrable g ⊢ False ** exact hf (hg.congr h.symm) ** Qed
| |
MeasureTheory.withDensityᵥ_toReal ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ≥0∞ hfm : AEMeasurable f hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ ⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f) ** have hfi := integrable_toReal_of_lintegral_ne_top hfm hf ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ≥0∞ hfm : AEMeasurable f hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hfi : Integrable fun x => ENNReal.toReal (f x) ⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f) ** haveI := isFiniteMeasure_withDensity hf ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ≥0∞ hfm : AEMeasurable f hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hfi : Integrable fun x => ENNReal.toReal (f x) this : IsFiniteMeasure (withDensity μ fun a => f a) ⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f) ** ext i hi ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ≥0∞ hfm : AEMeasurable f hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hfi : Integrable fun x => ENNReal.toReal (f x) this : IsFiniteMeasure (withDensity μ fun a => f a) i : Set α hi : MeasurableSet i ⊢ ↑(withDensityᵥ μ fun x => ENNReal.toReal (f x)) i = ↑(toSignedMeasure (withDensity μ f)) i ** rw [withDensityᵥ_apply hfi hi, toSignedMeasure_apply_measurable hi, withDensity_apply _ hi,
integral_toReal hfm.restrict] ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ≥0∞ hfm : AEMeasurable f hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hfi : Integrable fun x => ENNReal.toReal (f x) this : IsFiniteMeasure (withDensity μ fun a => f a) i : Set α hi : MeasurableSet i ⊢ ∀ᵐ (x : α) ∂restrict μ i, f x < ⊤ ** refine' ae_lt_top' hfm.restrict (ne_top_of_le_ne_top hf _) ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ≥0∞ hfm : AEMeasurable f hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hfi : Integrable fun x => ENNReal.toReal (f x) this : IsFiniteMeasure (withDensity μ fun a => f a) i : Set α hi : MeasurableSet i ⊢ ∫⁻ (x : α) in i, f x ∂μ ≤ ∫⁻ (x : α), f x ∂μ ** conv_rhs => rw [← set_lintegral_univ] ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ≥0∞ hfm : AEMeasurable f hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤ hfi : Integrable fun x => ENNReal.toReal (f x) this : IsFiniteMeasure (withDensity μ fun a => f a) i : Set α hi : MeasurableSet i ⊢ ∫⁻ (x : α) in i, f x ∂μ ≤ ∫⁻ (x : α) in Set.univ, f x ∂μ ** exact lintegral_mono_set (Set.subset_univ _) ** Qed
| |
MeasureTheory.withDensityᵥ_eq_withDensity_pos_part_sub_withDensity_neg_part ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ hfi : Integrable f ⊢ withDensityᵥ μ f = toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (-f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.2 ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ hfi : Integrable f this : IsFiniteMeasure (withDensity μ fun x => ENNReal.ofReal (f x)) ⊢ withDensityᵥ μ f = toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (-f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2 ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ hfi : Integrable f this✝ : IsFiniteMeasure (withDensity μ fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity μ fun x => ENNReal.ofReal ((-f) x)) ⊢ withDensityᵥ μ f = toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (-f x)) ** ext i hi ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E f : α → ℝ hfi : Integrable f this✝ : IsFiniteMeasure (withDensity μ fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity μ fun x => ENNReal.ofReal ((-f) x)) i : Set α hi : MeasurableSet i ⊢ ↑(withDensityᵥ μ f) i = ↑(toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity μ fun x => ENNReal.ofReal (-f x))) i ** rw [withDensityᵥ_apply hfi hi,
integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi.integrableOn,
VectorMeasure.sub_apply, toSignedMeasure_apply_measurable hi,
toSignedMeasure_apply_measurable hi, withDensity_apply _ hi, withDensity_apply _ hi] ** Qed
| |
MeasureTheory.Integrable.withDensityᵥ_trim_eq_integral ** α : Type u_1 β : Type u_2 m✝ : MeasurableSpace α μ✝ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : α → E m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ hf : Integrable f i : Set α hi : MeasurableSet i ⊢ ↑(VectorMeasure.trim (withDensityᵥ μ f) hm) i = ∫ (x : α) in i, f x ∂μ ** rw [VectorMeasure.trim_measurableSet_eq hm hi, withDensityᵥ_apply hf (hm _ hi)] ** Qed
| |
MeasureTheory.Integrable.withDensityᵥ_trim_absolutelyContinuous ** α : Type u_1 β : Type u_2 m✝ : MeasurableSpace α μ✝ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 hfi : Integrable f ⊢ VectorMeasure.trim (withDensityᵥ μ f) hm ≪ᵥ toENNRealVectorMeasure (Measure.trim μ hm) ** refine' VectorMeasure.AbsolutelyContinuous.mk fun j hj₁ hj₂ => _ ** α : Type u_1 β : Type u_2 m✝ : MeasurableSpace α μ✝ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 hfi : Integrable f j : Set α hj₁ : MeasurableSet j hj₂ : ↑(toENNRealVectorMeasure (Measure.trim μ hm)) j = 0 ⊢ ↑(VectorMeasure.trim (withDensityᵥ μ f) hm) j = 0 ** rw [Measure.toENNRealVectorMeasure_apply_measurable hj₁, trim_measurableSet_eq hm hj₁] at hj₂ ** α : Type u_1 β : Type u_2 m✝ : MeasurableSpace α μ✝ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 hfi : Integrable f j : Set α hj₁ : MeasurableSet j hj₂ : ↑↑μ j = 0 ⊢ ↑(VectorMeasure.trim (withDensityᵥ μ f) hm) j = 0 ** rw [VectorMeasure.trim_measurableSet_eq hm hj₁, withDensityᵥ_apply hfi (hm _ hj₁)] ** α : Type u_1 β : Type u_2 m✝ : MeasurableSpace α μ✝ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 hfi : Integrable f j : Set α hj₁ : MeasurableSet j hj₂ : ↑↑μ j = 0 ⊢ ∫ (x : α) in j, f x ∂μ = 0 ** simp only [Measure.restrict_eq_zero.mpr hj₂, integral_zero_measure] ** Qed
| |
MeasureTheory.withDensity_congr_ae ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f g : α → ℝ≥0∞ h : f =ᶠ[ae μ] g ⊢ withDensity μ f = withDensity μ g ** refine Measure.ext fun s hs => ?_ ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f g : α → ℝ≥0∞ h : f =ᶠ[ae μ] g s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ f) s = ↑↑(withDensity μ g) s ** rw [withDensity_apply _ hs, withDensity_apply _ hs] ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f g : α → ℝ≥0∞ h : f =ᶠ[ae μ] g s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ ** exact lintegral_congr_ae (ae_restrict_of_ae h) ** Qed
| |
MeasureTheory.withDensity_add_left ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ hf : Measurable f g : α → ℝ≥0∞ ⊢ withDensity μ (f + g) = withDensity μ f + withDensity μ g ** refine' Measure.ext fun s hs => _ ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ hf : Measurable f g : α → ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ (f + g)) s = ↑↑(withDensity μ f + withDensity μ g) s ** rw [withDensity_apply _ hs, Measure.add_apply, withDensity_apply _ hs, withDensity_apply _ hs,
← lintegral_add_left hf] ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ hf : Measurable f g : α → ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, (f + g) a ∂μ = ∫⁻ (a : α) in s, f a + g a ∂μ ** simp only [Pi.add_apply] ** Qed
| |
MeasureTheory.withDensity_add_right ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f g : α → ℝ≥0∞ hg : Measurable g ⊢ withDensity μ (f + g) = withDensity μ f + withDensity μ g ** simpa only [add_comm] using withDensity_add_left hg f ** Qed
| |
MeasureTheory.withDensity_add_measure ** α : Type u_1 m0 : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ ⊢ withDensity (μ + ν) f = withDensity μ f + withDensity ν f ** ext1 s hs ** case h α : Type u_1 m0 : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity (μ + ν) f) s = ↑↑(withDensity μ f + withDensity ν f) s ** simp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply] ** Qed
| |
MeasureTheory.withDensity_smul ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ withDensity μ (r • f) = r • withDensity μ f ** refine' Measure.ext fun s hs => _ ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ (r • f)) s = ↑↑(r • withDensity μ f) s ** rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf] ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ ** simp only [Pi.smul_apply, smul_eq_mul] ** Qed
| |
MeasureTheory.withDensity_smul' ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ ⊢ withDensity μ (r • f) = r • withDensity μ f ** refine' Measure.ext fun s hs => _ ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ (r • f)) s = ↑↑(r • withDensity μ f) s ** rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul' r f hr] ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ ** simp only [Pi.smul_apply, smul_eq_mul] ** Qed
| |
MeasureTheory.isFiniteMeasure_withDensity ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤ ⊢ ↑↑(withDensity μ f) univ < ⊤ ** rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, lt_top_iff_ne_top] ** Qed
| |
MeasureTheory.withDensity_absolutelyContinuous ** α : Type u_1 m0 : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ ⊢ withDensity μ f ≪ μ ** refine' AbsolutelyContinuous.mk fun s hs₁ hs₂ => _ ** α : Type u_1 m0 : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ s : Set α hs₁ : MeasurableSet s hs₂ : ↑↑μ s = 0 ⊢ ↑↑(withDensity μ f) s = 0 ** rw [withDensity_apply _ hs₁] ** α : Type u_1 m0 : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ s : Set α hs₁ : MeasurableSet s hs₂ : ↑↑μ s = 0 ⊢ ∫⁻ (a : α) in s, f a ∂μ = 0 ** exact set_lintegral_measure_zero _ _ hs₂ ** Qed
| |
MeasureTheory.withDensity_zero ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α ⊢ withDensity μ 0 = 0 ** ext1 s hs ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ 0) s = ↑↑0 s ** simp [withDensity_apply _ hs] ** Qed
| |
MeasureTheory.withDensity_one ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α ⊢ withDensity μ 1 = μ ** ext1 s hs ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ 1) s = ↑↑μ s ** simp [withDensity_apply _ hs] ** Qed
| |
MeasureTheory.withDensity_const ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α c : ℝ≥0∞ ⊢ (withDensity μ fun x => c) = c • μ ** ext1 s hs ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α c : ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ fun x => c) s = ↑↑(c • μ) s ** simp [withDensity_apply _ hs] ** Qed
| |
MeasureTheory.withDensity_tsum ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : ℕ → α → ℝ≥0∞ h : ∀ (i : ℕ), Measurable (f i) ⊢ withDensity μ (∑' (n : ℕ), f n) = sum fun n => withDensity μ (f n) ** ext1 s hs ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : ℕ → α → ℝ≥0∞ h : ∀ (i : ℕ), Measurable (f i) s : Set α hs : MeasurableSet s ⊢ ↑↑(withDensity μ (∑' (n : ℕ), f n)) s = ↑↑(sum fun n => withDensity μ (f n)) s ** simp_rw [sum_apply _ hs, withDensity_apply _ hs] ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : ℕ → α → ℝ≥0∞ h : ∀ (i : ℕ), Measurable (f i) s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, tsum (fun n => f n) a ∂μ = ∑' (i : ℕ), ∫⁻ (a : α) in s, f i a ∂μ ** change ∫⁻ x in s, (∑' n, f n) x ∂μ = ∑' i : ℕ, ∫⁻ x, f i x ∂μ.restrict s ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : ℕ → α → ℝ≥0∞ h : ∀ (i : ℕ), Measurable (f i) s : Set α hs : MeasurableSet s ⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∑' (i : ℕ), ∫⁻ (x : α) in s, f i x ∂μ ** rw [← lintegral_tsum fun i => (h i).aemeasurable] ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : ℕ → α → ℝ≥0∞ h : ∀ (i : ℕ), Measurable (f i) s : Set α hs : MeasurableSet s ⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ℕ), f i a ∂μ ** refine' lintegral_congr fun x => tsum_apply (Pi.summable.2 fun _ => ENNReal.summable) ** Qed
| |
MeasureTheory.withDensity_ofReal_mutuallySingular ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ hf : Measurable f ⊢ (withDensity μ fun x => ENNReal.ofReal (f x)) ⟂ₘ withDensity μ fun x => ENNReal.ofReal (-f x) ** set S : Set α := { x | f x < 0 } ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ hf : Measurable f S : Set α := {x | f x < 0} ⊢ (withDensity μ fun x => ENNReal.ofReal (f x)) ⟂ₘ withDensity μ fun x => ENNReal.ofReal (-f x) ** have hS : MeasurableSet S := measurableSet_lt hf measurable_const ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ hf : Measurable f S : Set α := {x | f x < 0} hS : MeasurableSet S ⊢ (withDensity μ fun x => ENNReal.ofReal (f x)) ⟂ₘ withDensity μ fun x => ENNReal.ofReal (-f x) ** refine' ⟨S, hS, _, _⟩ ** case refine'_1 α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ hf : Measurable f S : Set α := {x | f x < 0} hS : MeasurableSet S ⊢ ↑↑(withDensity μ fun x => ENNReal.ofReal (f x)) S = 0 ** rw [withDensity_apply _ hS, lintegral_eq_zero_iff hf.ennreal_ofReal, EventuallyEq] ** case refine'_1 α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ hf : Measurable f S : Set α := {x | f x < 0} hS : MeasurableSet S ⊢ ∀ᵐ (x : α) ∂restrict μ S, ENNReal.ofReal (f x) = OfNat.ofNat 0 x ** exact (ae_restrict_mem hS).mono fun x hx => ENNReal.ofReal_eq_zero.2 (le_of_lt hx) ** case refine'_2 α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ hf : Measurable f S : Set α := {x | f x < 0} hS : MeasurableSet S ⊢ ↑↑(withDensity μ fun x => ENNReal.ofReal (-f x)) Sᶜ = 0 ** rw [withDensity_apply _ hS.compl, lintegral_eq_zero_iff hf.neg.ennreal_ofReal, EventuallyEq] ** case refine'_2 α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ hf : Measurable f S : Set α := {x | f x < 0} hS : MeasurableSet S ⊢ ∀ᵐ (x : α) ∂restrict μ Sᶜ, ENNReal.ofReal (-f x) = OfNat.ofNat 0 x ** exact
(ae_restrict_mem hS.compl).mono fun x hx =>
ENNReal.ofReal_eq_zero.2 (not_lt.1 <| mt neg_pos.1 hx) ** Qed
| |
MeasureTheory.restrict_withDensity ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α s : Set α hs : MeasurableSet s f : α → ℝ≥0∞ ⊢ restrict (withDensity μ f) s = withDensity (restrict μ s) f ** ext1 t ht ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α s : Set α hs : MeasurableSet s f : α → ℝ≥0∞ t : Set α ht : MeasurableSet t ⊢ ↑↑(restrict (withDensity μ f) s) t = ↑↑(withDensity (restrict μ s) f) t ** rw [restrict_apply ht, withDensity_apply _ ht, withDensity_apply _ (ht.inter hs),
restrict_restrict ht] ** Qed
| |
MeasureTheory.withDensity_eq_zero ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h : withDensity μ f = 0 ⊢ f =ᶠ[ae μ] 0 ** rw [← lintegral_eq_zero_iff' hf, ← set_lintegral_univ, ← withDensity_apply _ MeasurableSet.univ,
h, Measure.coe_zero, Pi.zero_apply] ** Qed
| |
MeasureTheory.ae_withDensity_iff_ae_restrict ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α p : α → Prop f : α → ℝ≥0∞ hf : Measurable f ⊢ (∀ᵐ (x : α) ∂withDensity μ f, p x) ↔ ∀ᵐ (x : α) ∂restrict μ {x | f x ≠ 0}, p x ** rw [ae_withDensity_iff hf, ae_restrict_iff'] ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α p : α → Prop f : α → ℝ≥0∞ hf : Measurable f ⊢ (∀ᵐ (x : α) ∂μ, f x ≠ 0 → p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ {x | f x ≠ 0} → p x ** simp only [mem_setOf] ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α p : α → Prop f : α → ℝ≥0∞ hf : Measurable f ⊢ MeasurableSet {x | f x ≠ 0} ** exact hf (measurableSet_singleton 0).compl ** Qed
| |
MeasureTheory.aemeasurable_withDensity_ennreal_iff ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g : α → ℝ≥0∞ ⊢ AEMeasurable g ↔ AEMeasurable fun x => ↑(f x) * g x ** constructor ** case mp α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g : α → ℝ≥0∞ ⊢ AEMeasurable g → AEMeasurable fun x => ↑(f x) * g x ** rintro ⟨g', g'meas, hg'⟩ ** case mp.intro.intro α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' ⊢ AEMeasurable fun x => ↑(f x) * g x ** have A : MeasurableSet { x : α | f x ≠ 0 } := (hf (measurableSet_singleton 0)).compl ** case mp.intro.intro α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' A : MeasurableSet {x | f x ≠ 0} ⊢ AEMeasurable fun x => ↑(f x) * g x ** refine' ⟨fun x => f x * g' x, hf.coe_nnreal_ennreal.smul g'meas, _⟩ ** case mp.intro.intro α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' A : MeasurableSet {x | f x ≠ 0} ⊢ (fun x => ↑(f x) * g x) =ᶠ[ae μ] fun x => ↑(f x) * g' x ** apply ae_of_ae_restrict_of_ae_restrict_compl { x | f x ≠ 0 } ** case mp.intro.intro.ht α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂restrict μ {x | f x ≠ 0}, (fun x => ↑(f x) * g x) x = (fun x => ↑(f x) * g' x) x ** rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg' ** case mp.intro.intro.ht α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂restrict μ {x | f x ≠ 0}, (fun x => ↑(f x) * g x) x = (fun x => ↑(f x) * g' x) x ** rw [ae_restrict_iff' A] ** case mp.intro.intro.ht α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | f x ≠ 0} → (fun x => ↑(f x) * g x) x = (fun x => ↑(f x) * g' x) x ** filter_upwards [hg'] ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ (a : α), (↑(f a) ≠ 0 → g a = g' a) → f a ≠ 0 → ↑(f a) * g a = ↑(f a) * g' a ** intro a ha h'a ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} a : α ha : ↑(f a) ≠ 0 → g a = g' a h'a : f a ≠ 0 ⊢ ↑(f a) * g a = ↑(f a) * g' a ** have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne.def, coe_eq_zero] using h'a ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} a : α ha : ↑(f a) ≠ 0 → g a = g' a h'a : f a ≠ 0 this : ↑(f a) ≠ 0 ⊢ ↑(f a) * g a = ↑(f a) * g' a ** rw [ha this] ** α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x A : MeasurableSet {x | f x ≠ 0} a : α ha : ↑(f a) ≠ 0 → g a = g' a h'a : f a ≠ 0 ⊢ ↑(f a) ≠ 0 ** simpa only [Ne.def, coe_eq_zero] using h'a ** case mp.intro.intro.htc α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ᵐ (x : α) ∂restrict μ {x | f x ≠ 0}ᶜ, (fun x => ↑(f x) * g x) x = (fun x => ↑(f x) * g' x) x ** filter_upwards [ae_restrict_mem A.compl] ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' A : MeasurableSet {x | f x ≠ 0} ⊢ ∀ (a : α), a ∈ {x | f x ≠ 0}ᶜ → ↑(f a) * g a = ↑(f a) * g' a ** intro x hx ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' A : MeasurableSet {x | f x ≠ 0} x : α hx : x ∈ {x | f x ≠ 0}ᶜ ⊢ ↑(f x) * g x = ↑(f x) * g' x ** simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : g =ᶠ[ae (withDensity μ fun x => ↑(f x))] g' A : MeasurableSet {x | f x ≠ 0} x : α hx : f x = 0 ⊢ ↑(f x) * g x = ↑(f x) * g' x ** simp [hx] ** case mpr α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g : α → ℝ≥0∞ ⊢ (AEMeasurable fun x => ↑(f x) * g x) → AEMeasurable g ** rintro ⟨g', g'meas, hg'⟩ ** case mpr.intro.intro α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : (fun x => ↑(f x) * g x) =ᶠ[ae μ] g' ⊢ AEMeasurable g ** refine' ⟨fun x => ((f x)⁻¹ : ℝ≥0∞) * g' x, hf.coe_nnreal_ennreal.inv.smul g'meas, _⟩ ** case mpr.intro.intro α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : (fun x => ↑(f x) * g x) =ᶠ[ae μ] g' ⊢ g =ᶠ[ae (withDensity μ fun x => ↑(f x))] fun x => (↑(f x))⁻¹ * g' x ** rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] ** case mpr.intro.intro α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : (fun x => ↑(f x) * g x) =ᶠ[ae μ] g' ⊢ ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = (↑(f x))⁻¹ * g' x ** filter_upwards [hg'] ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : (fun x => ↑(f x) * g x) =ᶠ[ae μ] g' ⊢ ∀ (a : α), ↑(f a) * g a = g' a → ↑(f a) ≠ 0 → g a = (↑(f a))⁻¹ * g' a ** intro x hx h'x ** case h α : Type u_1 m0 : MeasurableSpace α μ : Measure α f : α → ℝ≥0 hf : Measurable f g g' : α → ℝ≥0∞ g'meas : Measurable g' hg' : (fun x => ↑(f x) * g x) =ᶠ[ae μ] g' x : α hx : ↑(f x) * g x = g' x h'x : ↑(f x) ≠ 0 ⊢ g x = (↑(f x))⁻¹ * g' x ** rw [← hx, ← mul_assoc, ENNReal.inv_mul_cancel h'x ENNReal.coe_ne_top, one_mul] ** Qed
| |
MeasureTheory.lintegral_withDensity_eq_lintegral_mul ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f ⊢ ∀ {g : α → ℝ≥0∞}, Measurable g → ∫⁻ (a : α), g a ∂withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ ** apply Measurable.ennreal_induction ** case h_ind α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f ⊢ ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → ∫⁻ (a : α), indicator s (fun x => c) a ∂withDensity μ f = ∫⁻ (a : α), (f * indicator s fun x => c) a ∂μ ** intro c s h_ms ** case h_ind α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f c : ℝ≥0∞ s : Set α h_ms : MeasurableSet s ⊢ ∫⁻ (a : α), indicator s (fun x => c) a ∂withDensity μ f = ∫⁻ (a : α), (f * indicator s fun x => c) a ∂μ ** simp [*, mul_comm _ c, ← indicator_mul_right] ** case h_add α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f ⊢ ∀ ⦃f_1 g : α → ℝ≥0∞⦄, Disjoint (Function.support f_1) (Function.support g) → Measurable f_1 → Measurable g → ∫⁻ (a : α), f_1 a ∂withDensity μ f = ∫⁻ (a : α), (f * f_1) a ∂μ → ∫⁻ (a : α), g a ∂withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ → ∫⁻ (a : α), (f_1 + g) a ∂withDensity μ f = ∫⁻ (a : α), (f * (f_1 + g)) a ∂μ ** intro g h _ h_mea_g _ h_ind_g h_ind_h ** case h_add α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f g h : α → ℝ≥0∞ a✝¹ : Disjoint (Function.support g) (Function.support h) h_mea_g : Measurable g a✝ : Measurable h h_ind_g : ∫⁻ (a : α), g a ∂withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ h_ind_h : ∫⁻ (a : α), h a ∂withDensity μ f = ∫⁻ (a : α), (f * h) a ∂μ ⊢ ∫⁻ (a : α), (g + h) a ∂withDensity μ f = ∫⁻ (a : α), (f * (g + h)) a ∂μ ** simp [mul_add, *, Measurable.mul] ** case h_iSup α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f ⊢ ∀ ⦃f_1 : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f_1 n)) → Monotone f_1 → (∀ (n : ℕ), ∫⁻ (a : α), f_1 n a ∂withDensity μ f = ∫⁻ (a : α), (f * f_1 n) a ∂μ) → ∫⁻ (a : α), (fun x => ⨆ n, f_1 n x) a ∂withDensity μ f = ∫⁻ (a : α), (f * fun x => ⨆ n, f_1 n x) a ∂μ ** intro g h_mea_g h_mono_g h_ind ** case h_iSup α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f g : ℕ → α → ℝ≥0∞ h_mea_g : ∀ (n : ℕ), Measurable (g n) h_mono_g : Monotone g h_ind : ∀ (n : ℕ), ∫⁻ (a : α), g n a ∂withDensity μ f = ∫⁻ (a : α), (f * g n) a ∂μ ⊢ ∫⁻ (a : α), (fun x => ⨆ n, g n x) a ∂withDensity μ f = ∫⁻ (a : α), (f * fun x => ⨆ n, g n x) a ∂μ ** have : Monotone fun n a => f a * g n a := fun m n hmn x => mul_le_mul_left' (h_mono_g hmn x) _ ** case h_iSup α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ h_mf : Measurable f g : ℕ → α → ℝ≥0∞ h_mea_g : ∀ (n : ℕ), Measurable (g n) h_mono_g : Monotone g h_ind : ∀ (n : ℕ), ∫⁻ (a : α), g n a ∂withDensity μ f = ∫⁻ (a : α), (f * g n) a ∂μ this : Monotone fun n a => f a * g n a ⊢ ∫⁻ (a : α), (fun x => ⨆ n, g n x) a ∂withDensity μ f = ∫⁻ (a : α), (f * fun x => ⨆ n, g n x) a ∂μ ** simp [lintegral_iSup, ENNReal.mul_iSup, h_mf.mul (h_mea_g _), *] ** Qed
| |
MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f g : α → ℝ≥0∞ hf : Measurable f hg : Measurable g s : Set α hs : MeasurableSet s ⊢ ∫⁻ (x : α) in s, g x ∂withDensity μ f = ∫⁻ (x : α) in s, (f * g) x ∂μ ** rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul _ hf hg] ** Qed
| |
MeasureTheory.lintegral_withDensity_le_lintegral_mul ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ f_meas : Measurable f g : α → ℝ≥0∞ ⊢ ∫⁻ (a : α), g a ∂withDensity μ f ≤ ∫⁻ (a : α), (f * g) a ∂μ ** rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral] ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ f_meas : Measurable f g : α → ℝ≥0∞ ⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => g a), ∫⁻ (a : α), g_1 a ∂withDensity μ f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => (f * g) a), ∫⁻ (a : α), g_1 a ∂μ ** refine' iSup₂_le fun i i_meas => iSup_le fun hi => _ ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ f_meas : Measurable f g i : α → ℝ≥0∞ i_meas : Measurable i hi : i ≤ fun a => g a ⊢ ∫⁻ (a : α), i a ∂withDensity μ f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => (f * g) a), ∫⁻ (a : α), g_1 a ∂μ ** have A : f * i ≤ f * g := fun x => mul_le_mul_left' (hi x) _ ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ f_meas : Measurable f g i : α → ℝ≥0∞ i_meas : Measurable i hi : i ≤ fun a => g a A : f * i ≤ f * g ⊢ ∫⁻ (a : α), i a ∂withDensity μ f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => (f * g) a), ∫⁻ (a : α), g_1 a ∂μ ** refine' le_iSup₂_of_le (f * i) (f_meas.mul i_meas) _ ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ f_meas : Measurable f g i : α → ℝ≥0∞ i_meas : Measurable i hi : i ≤ fun a => g a A : f * i ≤ f * g ⊢ ∫⁻ (a : α), i a ∂withDensity μ f ≤ ⨆ (_ : f * i ≤ fun a => (f * g) a), ∫⁻ (a : α), (f * i) a ∂μ ** exact le_iSup_of_le A (le_of_eq (lintegral_withDensity_eq_lintegral_mul _ f_meas i_meas)) ** Qed
| |
MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ f_meas : Measurable f g : α → ℝ≥0∞ s : Set α hs : MeasurableSet s hf : ∀ᵐ (x : α) ∂Measure.restrict μ s, f x < ⊤ ⊢ ∫⁻ (a : α) in s, g a ∂withDensity μ f = ∫⁻ (a : α) in s, (f * g) a ∂μ ** rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable _ f_meas hf] ** Qed
| |
MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable₀ ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ ⊢ ∫⁻ (a : α), g a ∂withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ ** let f' := hf.mk f ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ ∫⁻ (a : α), g a ∂withDensity μ f = ∫⁻ (a : α), (f * g) a ∂μ ** calc
∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, g a ∂μ.withDensity f' := by
rw [withDensity_congr_ae hf.ae_eq_mk]
_ = ∫⁻ a, (f' * g) a ∂μ := by
apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk
filter_upwards [h'f, hf.ae_eq_mk]
intro x hx h'x
rwa [← h'x]
_ = ∫⁻ a, (f * g) a ∂μ := by
apply lintegral_congr_ae
filter_upwards [hf.ae_eq_mk]
intro x hx
simp only [hx, Pi.mul_apply] ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ ∫⁻ (a : α), g a ∂withDensity μ f = ∫⁻ (a : α), g a ∂withDensity μ f' ** rw [withDensity_congr_ae hf.ae_eq_mk] ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ ∫⁻ (a : α), g a ∂withDensity μ f' = ∫⁻ (a : α), (f' * g) a ∂μ ** apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk ** case hf α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ ∀ᵐ (x : α) ∂μ, AEMeasurable.mk f hf x < ⊤ ** filter_upwards [h'f, hf.ae_eq_mk] ** case h α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ ∀ (a : α), f a < ⊤ → f a = AEMeasurable.mk f hf a → AEMeasurable.mk f hf a < ⊤ ** intro x hx h'x ** case h α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf x : α hx : f x < ⊤ h'x : f x = AEMeasurable.mk f hf x ⊢ AEMeasurable.mk f hf x < ⊤ ** rwa [← h'x] ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ ∫⁻ (a : α), (f' * g) a ∂μ = ∫⁻ (a : α), (f * g) a ∂μ ** apply lintegral_congr_ae ** case h α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ (fun a => (f' * g) a) =ᶠ[ae μ] fun a => (f * g) a ** filter_upwards [hf.ae_eq_mk] ** case h α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf ⊢ ∀ (a : α), f a = AEMeasurable.mk f hf a → (f' * g) a = (f * g) a ** intro x hx ** case h α : Type u_1 m0 : MeasurableSpace α μ✝ μ : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤ g : α → ℝ≥0∞ f' : α → ℝ≥0∞ := AEMeasurable.mk f hf x : α hx : f x = AEMeasurable.mk f hf x ⊢ (f' * g) x = (f * g) x ** simp only [hx, Pi.mul_apply] ** Qed
| |
IsUnifLocDoublingMeasure.eventually_measure_mul_le_scalingConstantOf_mul ** α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ ⊢ ∃ R, 0 < R ∧ ∀ (x : α) (t r : ℝ), t ∈ Ioc 0 K → r ≤ R → ↑↑μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** have h := Classical.choose_spec (exists_eventually_forall_measure_closedBall_le_mul μ K) ** α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) ⊢ ∃ R, 0 < R ∧ ∀ (x : α) (t r : ℝ), t ∈ Ioc 0 K → r ≤ R → ↑↑μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** rcases mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 h with ⟨R, Rpos, hR⟩ ** case intro.intro α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} ⊢ ∃ R, 0 < R ∧ ∀ (x : α) (t r : ℝ), t ∈ Ioc 0 K → r ≤ R → ↑↑μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** refine' ⟨R, Rpos, fun x t r ht hr => _⟩ ** case intro.intro α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t r : ℝ ht : t ∈ Ioc 0 K hr : r ≤ R ⊢ ↑↑μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** rcases lt_trichotomy r 0 with (rneg | rfl | rpos) ** case intro.intro.inl α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t r : ℝ ht : t ∈ Ioc 0 K hr : r ≤ R rneg : r < 0 ⊢ ↑↑μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** have : t * r < 0 := mul_neg_of_pos_of_neg ht.1 rneg ** case intro.intro.inl α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t r : ℝ ht : t ∈ Ioc 0 K hr : r ≤ R rneg : r < 0 this : t * r < 0 ⊢ ↑↑μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** simp only [closedBall_eq_empty.2 this, measure_empty, zero_le'] ** case intro.intro.inr.inl α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t : ℝ ht : t ∈ Ioc 0 K hr : 0 ≤ R ⊢ ↑↑μ (closedBall x (t * 0)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x 0) ** simp only [mul_zero, closedBall_zero] ** case intro.intro.inr.inl α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t : ℝ ht : t ∈ Ioc 0 K hr : 0 ≤ R ⊢ ↑↑μ {x} ≤ ↑(scalingConstantOf μ K) * ↑↑μ {x} ** refine' le_mul_of_one_le_of_le _ le_rfl ** case intro.intro.inr.inl α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t : ℝ ht : t ∈ Ioc 0 K hr : 0 ≤ R ⊢ 1 ≤ ↑(scalingConstantOf μ K) ** apply ENNReal.one_le_coe_iff.2 (le_max_right _ _) ** case intro.intro.inr.inr α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t r : ℝ ht : t ∈ Ioc 0 K hr : r ≤ R rpos : 0 < r ⊢ ↑↑μ (closedBall x (t * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** apply (hR ⟨rpos, hr⟩ x t ht.2).trans _ ** α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ h : ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε) R : ℝ Rpos : R ∈ Ioi 0 hR : Ioc 0 R ⊆ {x | (fun ε => ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x ε)) x} x : α t r : ℝ ht : t ∈ Ioc 0 K hr : r ≤ R rpos : 0 < r ⊢ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x r) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** exact mul_le_mul_right' (ENNReal.coe_le_coe.2 (le_max_left _ _)) _ ** Qed
| |
IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul ** α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K : ℝ ⊢ ∀ᶠ (r : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α), ↑↑μ (closedBall x (K * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** filter_upwards [Classical.choose_spec
(exists_eventually_forall_measure_closedBall_le_mul μ K)] with r hr x ** case h α : Type u_1 inst✝² : MetricSpace α inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : IsUnifLocDoublingMeasure μ K r : ℝ hr : ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * r)) ≤ ↑(Classical.choose (_ : ∃ C, ∀ᶠ (ε : ℝ) in 𝓝[Ioi 0] 0, ∀ (x : α) (t : ℝ), t ≤ K → ↑↑μ (closedBall x (t * ε)) ≤ ↑C * ↑↑μ (closedBall x ε))) * ↑↑μ (closedBall x r) x : α ⊢ ↑↑μ (closedBall x (K * r)) ≤ ↑(scalingConstantOf μ K) * ↑↑μ (closedBall x r) ** exact (hr x K le_rfl).trans (mul_le_mul_right' (ENNReal.coe_le_coe.2 (le_max_left _ _)) _) ** Qed
| |
IsOpen.measure_eq_zero_iff ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X hU : IsOpen U ⊢ ↑↑μ U = 0 ↔ U = ∅ ** simpa only [not_lt, nonpos_iff_eq_zero, not_nonempty_iff_eq_empty] using
not_congr (hU.measure_pos_iff μ) ** Qed
| |
IsOpen.measure_zero_iff_eq_empty ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X hU : IsOpen U h : U = ∅ ⊢ ↑↑μ U = 0 ** simp [h] ** Qed
| |
IsOpen.ae_eq_empty_iff_eq ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X hU : IsOpen U ⊢ U =ᶠ[ae μ] ∅ ↔ U = ∅ ** rw [ae_eq_empty, hU.measure_zero_iff_eq_empty] ** Qed
| |
IsClosed.ae_eq_univ_iff_eq ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X hF : IsClosed F ⊢ F =ᶠ[ae μ] univ ↔ F = univ ** refine' ⟨fun h ↦ _, fun h ↦ by rw [h]⟩ ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X hF : IsClosed F h : F =ᶠ[ae μ] univ ⊢ F = univ ** rwa [ae_eq_univ, hF.isOpen_compl.measure_eq_zero_iff μ, compl_empty_iff] at h ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X hF : IsClosed F h : F = univ ⊢ F =ᶠ[ae μ] univ ** rw [h] ** Qed
| |
IsClosed.measure_eq_univ_iff_eq ** X : Type u_1 Y : Type u_2 inst✝⁵ : TopologicalSpace X m : MeasurableSpace X inst✝⁴ : TopologicalSpace Y inst✝³ : T2Space Y μ ν : Measure X inst✝² : IsOpenPosMeasure μ s U F : Set X x : X inst✝¹ : OpensMeasurableSpace X inst✝ : IsFiniteMeasure μ hF : IsClosed F ⊢ ↑↑μ F = ↑↑μ univ ↔ F = univ ** rw [← ae_eq_univ_iff_measure_eq hF.measurableSet.nullMeasurableSet, hF.ae_eq_univ_iff_eq] ** Qed
| |
IsClosed.measure_eq_one_iff_eq_univ ** X : Type u_1 Y : Type u_2 inst✝⁵ : TopologicalSpace X m : MeasurableSpace X inst✝⁴ : TopologicalSpace Y inst✝³ : T2Space Y μ ν : Measure X inst✝² : IsOpenPosMeasure μ s U F : Set X x : X inst✝¹ : OpensMeasurableSpace X inst✝ : IsProbabilityMeasure μ hF : IsClosed F ⊢ ↑↑μ F = 1 ↔ F = univ ** rw [← measure_univ (μ := μ), hF.measure_eq_univ_iff_eq] ** Qed
| |
MeasureTheory.Measure.eqOn_open_of_ae_eq ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y h : f =ᶠ[ae (restrict μ U)] g hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U ⊢ EqOn f g U ** replace h := ae_imp_of_ae_restrict h ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : ∀ᵐ (x : X) ∂μ, x ∈ U → f x = g x ⊢ EqOn f g U ** simp only [EventuallyEq, ae_iff, not_imp] at h ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : ↑↑μ {a | a ∈ U ∧ ¬f a = g a} = 0 ⊢ EqOn f g U ** have : IsOpen (U ∩ { a | f a ≠ g a }) := by
refine' isOpen_iff_mem_nhds.mpr fun a ha => inter_mem (hU.mem_nhds ha.1) _
rcases ha with ⟨ha : a ∈ U, ha' : (f a, g a) ∈ (diagonal Y)ᶜ⟩
exact
(hf.continuousAt (hU.mem_nhds ha)).prod_mk_nhds (hg.continuousAt (hU.mem_nhds ha))
(isClosed_diagonal.isOpen_compl.mem_nhds ha') ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : ↑↑μ {a | a ∈ U ∧ ¬f a = g a} = 0 this : IsOpen (U ∩ {a | f a ≠ g a}) ⊢ EqOn f g U ** replace := (this.eq_empty_of_measure_zero h).le ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : ↑↑μ {a | a ∈ U ∧ ¬f a = g a} = 0 this : U ∩ {a | f a ≠ g a} ≤ ∅ ⊢ EqOn f g U ** exact fun x hx => Classical.not_not.1 fun h => this ⟨hx, h⟩ ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : ↑↑μ {a | a ∈ U ∧ ¬f a = g a} = 0 ⊢ IsOpen (U ∩ {a | f a ≠ g a}) ** refine' isOpen_iff_mem_nhds.mpr fun a ha => inter_mem (hU.mem_nhds ha.1) _ ** X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : ↑↑μ {a | a ∈ U ∧ ¬f a = g a} = 0 a : X ha : a ∈ U ∩ {a | f a ≠ g a} ⊢ {a | f a ≠ g a} ∈ 𝓝 a ** rcases ha with ⟨ha : a ∈ U, ha' : (f a, g a) ∈ (diagonal Y)ᶜ⟩ ** case intro X : Type u_1 Y : Type u_2 inst✝³ : TopologicalSpace X m : MeasurableSpace X inst✝² : TopologicalSpace Y inst✝¹ : T2Space Y μ ν : Measure X inst✝ : IsOpenPosMeasure μ s U F : Set X x : X f g : X → Y hU : IsOpen U hf : ContinuousOn f U hg : ContinuousOn g U h : ↑↑μ {a | a ∈ U ∧ ¬f a = g a} = 0 a : X ha : a ∈ U ha' : (f a, g a) ∈ (diagonal Y)ᶜ ⊢ {a | f a ≠ g a} ∈ 𝓝 a ** exact
(hf.continuousAt (hU.mem_nhds ha)).prod_mk_nhds (hg.continuousAt (hU.mem_nhds ha))
(isClosed_diagonal.isOpen_compl.mem_nhds ha') ** Qed
| |
MeasureTheory.Measure.InnerRegular.measure_eq_iSup ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q hU : q U ⊢ ↑↑μ U = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), ↑↑μ K ** refine'
le_antisymm (le_of_forall_lt fun r hr => _) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK) ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q hU : q U r : ℝ≥0∞ hr : r < ↑↑μ U ⊢ r < ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), ↑↑μ K ** simpa only [lt_iSup_iff, exists_prop] using H hU r hr ** Qed
| |
MeasureTheory.Measure.InnerRegular.exists_subset_lt_add ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q h0 : p ∅ hU : q U hμU : ↑↑μ U ≠ ⊤ hε : ε ≠ 0 ⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε ** cases' eq_or_ne (μ U) 0 with h₀ h₀ ** case inl α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q h0 : p ∅ hU : q U hμU : ↑↑μ U ≠ ⊤ hε : ε ≠ 0 h₀ : ↑↑μ U = 0 ⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε ** refine' ⟨∅, empty_subset _, h0, _⟩ ** case inl α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q h0 : p ∅ hU : q U hμU : ↑↑μ U ≠ ⊤ hε : ε ≠ 0 h₀ : ↑↑μ U = 0 ⊢ ↑↑μ U < ↑↑μ ∅ + ε ** rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero] ** case inr α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q h0 : p ∅ hU : q U hμU : ↑↑μ U ≠ ⊤ hε : ε ≠ 0 h₀ : ↑↑μ U ≠ 0 ⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε ** rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩ ** case inr.intro.intro.intro α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q h0 : p ∅ hU : q U hμU : ↑↑μ U ≠ ⊤ hε : ε ≠ 0 h₀ : ↑↑μ U ≠ 0 K : Set α hKU : K ⊆ U hKc : p K hrK : ↑↑μ U - ε < ↑↑μ K ⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε ** exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩ ** Qed
| |
MeasureTheory.Measure.InnerRegular.map ** α✝ : Type u_1 m : MeasurableSpace α✝ μ✝ : Measure α✝ p q : Set α✝ → Prop U : Set α✝ ε : ℝ≥0∞ α : Type u_2 β : Type u_3 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure α pa qa : Set α → Prop H : InnerRegular μ pa qa f : α ≃ β hf : AEMeasurable ↑f pb qb : Set β → Prop hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U) hAB' : ∀ (K : Set α), pa K → pb (↑f '' K) hB₁ : ∀ (K : Set β), pb K → MeasurableSet K hB₂ : ∀ (U : Set β), qb U → MeasurableSet U ⊢ InnerRegular (Measure.map (↑f) μ) pb qb ** intro U hU r hr ** α✝ : Type u_1 m : MeasurableSpace α✝ μ✝ : Measure α✝ p q : Set α✝ → Prop U✝ : Set α✝ ε : ℝ≥0∞ α : Type u_2 β : Type u_3 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure α pa qa : Set α → Prop H : InnerRegular μ pa qa f : α ≃ β hf : AEMeasurable ↑f pb qb : Set β → Prop hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U) hAB' : ∀ (K : Set α), pa K → pb (↑f '' K) hB₁ : ∀ (K : Set β), pb K → MeasurableSet K hB₂ : ∀ (U : Set β), qb U → MeasurableSet U U : Set β hU : qb U r : ℝ≥0∞ hr : r < ↑↑(Measure.map (↑f) μ) U ⊢ ∃ K, K ⊆ U ∧ pb K ∧ r < ↑↑(Measure.map (↑f) μ) K ** rw [map_apply_of_aemeasurable hf (hB₂ _ hU)] at hr ** α✝ : Type u_1 m : MeasurableSpace α✝ μ✝ : Measure α✝ p q : Set α✝ → Prop U✝ : Set α✝ ε : ℝ≥0∞ α : Type u_2 β : Type u_3 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure α pa qa : Set α → Prop H : InnerRegular μ pa qa f : α ≃ β hf : AEMeasurable ↑f pb qb : Set β → Prop hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U) hAB' : ∀ (K : Set α), pa K → pb (↑f '' K) hB₁ : ∀ (K : Set β), pb K → MeasurableSet K hB₂ : ∀ (U : Set β), qb U → MeasurableSet U U : Set β hU : qb U r : ℝ≥0∞ hr : r < ↑↑μ (↑f ⁻¹' U) ⊢ ∃ K, K ⊆ U ∧ pb K ∧ r < ↑↑(Measure.map (↑f) μ) K ** rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩ ** case intro.intro.intro α✝ : Type u_1 m : MeasurableSpace α✝ μ✝ : Measure α✝ p q : Set α✝ → Prop U✝ : Set α✝ ε : ℝ≥0∞ α : Type u_2 β : Type u_3 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure α pa qa : Set α → Prop H : InnerRegular μ pa qa f : α ≃ β hf : AEMeasurable ↑f pb qb : Set β → Prop hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U) hAB' : ∀ (K : Set α), pa K → pb (↑f '' K) hB₁ : ∀ (K : Set β), pb K → MeasurableSet K hB₂ : ∀ (U : Set β), qb U → MeasurableSet U U : Set β hU : qb U r : ℝ≥0∞ hr : r < ↑↑μ (↑f ⁻¹' U) K : Set α hKU : K ⊆ ↑f ⁻¹' U hKc : pa K hK : r < ↑↑μ K ⊢ ∃ K, K ⊆ U ∧ pb K ∧ r < ↑↑(Measure.map (↑f) μ) K ** refine' ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, _⟩ ** case intro.intro.intro α✝ : Type u_1 m : MeasurableSpace α✝ μ✝ : Measure α✝ p q : Set α✝ → Prop U✝ : Set α✝ ε : ℝ≥0∞ α : Type u_2 β : Type u_3 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β μ : Measure α pa qa : Set α → Prop H : InnerRegular μ pa qa f : α ≃ β hf : AEMeasurable ↑f pb qb : Set β → Prop hAB : ∀ (U : Set β), qb U → qa (↑f ⁻¹' U) hAB' : ∀ (K : Set α), pa K → pb (↑f '' K) hB₁ : ∀ (K : Set β), pb K → MeasurableSet K hB₂ : ∀ (U : Set β), qb U → MeasurableSet U U : Set β hU : qb U r : ℝ≥0∞ hr : r < ↑↑μ (↑f ⁻¹' U) K : Set α hKU : K ⊆ ↑f ⁻¹' U hKc : pa K hK : r < ↑↑μ K ⊢ r < ↑↑(Measure.map (↑f) μ) (↑f '' K) ** rwa [map_apply_of_aemeasurable hf (hB₁ _ <| hAB' _ hKc), f.preimage_image] ** Qed
| |
MeasureTheory.Measure.InnerRegular.smul ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ H : InnerRegular μ p q c : ℝ≥0∞ ⊢ InnerRegular (c • μ) p q ** intro U hU r hr ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U✝ : Set α ε : ℝ≥0∞ H : InnerRegular μ p q c : ℝ≥0∞ U : Set α hU : q U r : ℝ≥0∞ hr : r < ↑↑(c • μ) U ⊢ ∃ K, K ⊆ U ∧ p K ∧ r < ↑↑(c • μ) K ** rw [smul_apply, H.measure_eq_iSup hU, smul_eq_mul] at hr ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U✝ : Set α ε : ℝ≥0∞ H : InnerRegular μ p q c : ℝ≥0∞ U : Set α hU : q U r : ℝ≥0∞ hr : r < c * ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), ↑↑μ K ⊢ ∃ K, K ⊆ U ∧ p K ∧ r < ↑↑(c • μ) K ** simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr ** Qed
| |
MeasureTheory.Measure.InnerRegular.trans ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U : Set α ε : ℝ≥0∞ q' : Set α → Prop H : InnerRegular μ p q H' : InnerRegular μ q q' ⊢ InnerRegular μ p q' ** intro U hU r hr ** α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U✝ : Set α ε : ℝ≥0∞ q' : Set α → Prop H : InnerRegular μ p q H' : InnerRegular μ q q' U : Set α hU : q' U r : ℝ≥0∞ hr : r < ↑↑μ U ⊢ ∃ K, K ⊆ U ∧ p K ∧ r < ↑↑μ K ** rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩ ** case intro.intro.intro α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U✝ : Set α ε : ℝ≥0∞ q' : Set α → Prop H : InnerRegular μ p q H' : InnerRegular μ q q' U : Set α hU : q' U r : ℝ≥0∞ hr : r < ↑↑μ U F : Set α hFU : F ⊆ U hqF : q F hF : r < ↑↑μ F ⊢ ∃ K, K ⊆ U ∧ p K ∧ r < ↑↑μ K ** rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩ ** case intro.intro.intro.intro.intro.intro α : Type u_1 m : MeasurableSpace α μ : Measure α p q : Set α → Prop U✝ : Set α ε : ℝ≥0∞ q' : Set α → Prop H : InnerRegular μ p q H' : InnerRegular μ q q' U : Set α hU : q' U r : ℝ≥0∞ hr : r < ↑↑μ U F : Set α hFU : F ⊆ U hqF : q F hF : r < ↑↑μ F K : Set α hKF : K ⊆ F hpK : p K hrK : r < ↑↑μ K ⊢ ∃ K, K ⊆ U ∧ p K ∧ r < ↑↑μ K ** exact ⟨K, hKF.trans hFU, hpK, hrK⟩ ** Qed
| |
Set.measure_eq_iInf_isOpen ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ ⊢ ↑↑μ A = ⨅ U, ⨅ (_ : A ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U ** refine' le_antisymm (le_iInf₂ fun s hs => le_iInf fun _ => μ.mono hs) _ ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ ⊢ ⨅ U, ⨅ (_ : A ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U ≤ ↑↑μ A ** refine' le_of_forall_lt' fun r hr => _ ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ r : ℝ≥0∞ hr : ↑↑μ A < r ⊢ ⨅ U, ⨅ (_ : A ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ U < r ** simpa only [iInf_lt_iff, exists_prop] using A.exists_isOpen_lt_of_lt r hr ** Qed
| |
Set.exists_isOpen_le_add ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U ≤ ↑↑μ A + ε ** rcases eq_or_ne (μ A) ∞ with (H | H) ** case inl α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ ε : ℝ≥0∞ hε : ε ≠ 0 H : ↑↑μ A = ⊤ ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U ≤ ↑↑μ A + ε ** exact ⟨univ, subset_univ _, isOpen_univ, by simp only [H, _root_.top_add, le_top]⟩ ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ ε : ℝ≥0∞ hε : ε ≠ 0 H : ↑↑μ A = ⊤ ⊢ ↑↑μ univ ≤ ↑↑μ A + ε ** simp only [H, _root_.top_add, le_top] ** case inr α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ ε : ℝ≥0∞ hε : ε ≠ 0 H : ↑↑μ A ≠ ⊤ ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U ≤ ↑↑μ A + ε ** rcases A.exists_isOpen_lt_add H hε with ⟨U, AU, U_open, hU⟩ ** case inr.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α A : Set α μ : Measure α inst✝ : OuterRegular μ ε : ℝ≥0∞ hε : ε ≠ 0 H : ↑↑μ A ≠ ⊤ U : Set α AU : U ⊇ A U_open : IsOpen U hU : ↑↑μ U < ↑↑μ A + ε ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U ≤ ↑↑μ A + ε ** exact ⟨U, AU, U_open, hU.le⟩ ** Qed
| |
MeasurableSet.exists_isOpen_diff_lt ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ : Measure α inst✝ : OuterRegular μ A : Set α hA : MeasurableSet A hA' : ↑↑μ A ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < ⊤ ∧ ↑↑μ (U \ A) < ε ** rcases A.exists_isOpen_lt_add hA' hε with ⟨U, hAU, hUo, hU⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ : Measure α inst✝ : OuterRegular μ A : Set α hA : MeasurableSet A hA' : ↑↑μ A ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 U : Set α hAU : U ⊇ A hUo : IsOpen U hU : ↑↑μ U < ↑↑μ A + ε ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < ⊤ ∧ ↑↑μ (U \ A) < ε ** use U, hAU, hUo, hU.trans_le le_top ** case right α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ : Measure α inst✝ : OuterRegular μ A : Set α hA : MeasurableSet A hA' : ↑↑μ A ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 U : Set α hAU : U ⊇ A hUo : IsOpen U hU : ↑↑μ U < ↑↑μ A + ε ⊢ ↑↑μ (U \ A) < ε ** exact measure_diff_lt_of_lt_add hA hAU hA' hU ** Qed
| |
MeasureTheory.Measure.OuterRegular.map ** α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace α μ✝ : Measure α inst✝⁴ : OpensMeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : TopologicalSpace β inst✝¹ : BorelSpace β f : α ≃ₜ β μ : Measure α inst✝ : OuterRegular μ ⊢ OuterRegular (map (↑f) μ) ** refine' ⟨fun A hA r hr => _⟩ ** α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace α μ✝ : Measure α inst✝⁴ : OpensMeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : TopologicalSpace β inst✝¹ : BorelSpace β f : α ≃ₜ β μ : Measure α inst✝ : OuterRegular μ A : Set β hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑(map (↑f) μ) A ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑(map (↑f) μ) U < r ** rw [map_apply f.measurable hA, ← f.image_symm] at hr ** α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace α μ✝ : Measure α inst✝⁴ : OpensMeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : TopologicalSpace β inst✝¹ : BorelSpace β f : α ≃ₜ β μ : Measure α inst✝ : OuterRegular μ A : Set β hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ (↑(Homeomorph.symm f) '' A) ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑(map (↑f) μ) U < r ** rcases Set.exists_isOpen_lt_of_lt _ r hr with ⟨U, hAU, hUo, hU⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace α μ✝ : Measure α inst✝⁴ : OpensMeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : TopologicalSpace β inst✝¹ : BorelSpace β f : α ≃ₜ β μ : Measure α inst✝ : OuterRegular μ A : Set β hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ (↑(Homeomorph.symm f) '' A) U : Set α hAU : U ⊇ ↑(Homeomorph.symm f) '' A hUo : IsOpen U hU : ↑↑μ U < r ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑(map (↑f) μ) U < r ** have : IsOpen (f.symm ⁻¹' U) := hUo.preimage f.symm.continuous ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace α μ✝ : Measure α inst✝⁴ : OpensMeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : TopologicalSpace β inst✝¹ : BorelSpace β f : α ≃ₜ β μ : Measure α inst✝ : OuterRegular μ A : Set β hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ (↑(Homeomorph.symm f) '' A) U : Set α hAU : U ⊇ ↑(Homeomorph.symm f) '' A hUo : IsOpen U hU : ↑↑μ U < r this : IsOpen (↑(Homeomorph.symm f) ⁻¹' U) ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑(map (↑f) μ) U < r ** refine' ⟨f.symm ⁻¹' U, image_subset_iff.1 hAU, this, _⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝⁶ : MeasurableSpace α inst✝⁵ : TopologicalSpace α μ✝ : Measure α inst✝⁴ : OpensMeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : TopologicalSpace β inst✝¹ : BorelSpace β f : α ≃ₜ β μ : Measure α inst✝ : OuterRegular μ A : Set β hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ (↑(Homeomorph.symm f) '' A) U : Set α hAU : U ⊇ ↑(Homeomorph.symm f) '' A hUo : IsOpen U hU : ↑↑μ U < r this : IsOpen (↑(Homeomorph.symm f) ⁻¹' U) ⊢ ↑↑(map (↑f) μ) (↑(Homeomorph.symm f) ⁻¹' U) < r ** rwa [map_apply f.measurable this.measurableSet, f.preimage_symm, f.preimage_image] ** Qed
| |
MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} ⊢ OuterRegular μ ** refine' ⟨fun A hA r hr => _⟩ ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} A : Set α hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ A ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r ** have hm : ∀ n, MeasurableSet (s.set n) := fun n => (s.set_mem n).1.measurableSet ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} A : Set α hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ A hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r ** haveI : ∀ n, OuterRegular (μ.restrict (s.set n)) := fun n => (s.set_mem n).2 ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} A : Set α hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ A hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) ⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r ** obtain ⟨A, hAm, hAs, hAd, rfl⟩ :
∃ A' : ℕ → Set α,
(∀ n, MeasurableSet (A' n)) ∧
(∀ n, A' n ⊆ s.set n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ n, A' n := by
refine'
⟨fun n => A ∩ disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
(inter_subset_right _ _).trans (disjointed_subset _ _),
(disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
rw [← inter_iUnion, iUnion_disjointed, s.spanning, inter_univ] ** case intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) ⊢ ∃ U, U ⊇ ⋃ n, A n ∧ IsOpen U ∧ ↑↑μ U < r ** rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩ ** case intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) ⊢ ∃ U, U ⊇ ⋃ n, A n ∧ IsOpen U ∧ ↑↑μ U < r ** rw [lt_tsub_iff_right, add_comm] at hδε ** case intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r ⊢ ∃ U, U ⊇ ⋃ n, A n ∧ IsOpen U ∧ ↑↑μ U < r ** have : ∀ n, ∃ (U : _) (_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n := by
intro n
have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n)
have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
rw [H₁]
exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne
rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩ ** case intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this✝ : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r this : ∀ (n : ℕ), ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n ⊢ ∃ U, U ⊇ ⋃ n, A n ∧ IsOpen U ∧ ↑↑μ U < r ** choose U hAU hUo hU using this ** case intro.intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r U : ℕ → Set α hAU : ∀ (n : ℕ), U n ⊇ A n hUo : ∀ (n : ℕ), IsOpen (U n) hU : ∀ (n : ℕ), ↑↑μ (U n) < ↑↑μ (A n) + δ n ⊢ ∃ U, U ⊇ ⋃ n, A n ∧ IsOpen U ∧ ↑↑μ U < r ** refine' ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, _⟩ ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} A : Set α hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ A hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) ⊢ ∃ A', (∀ (n : ℕ), MeasurableSet (A' n)) ∧ (∀ (n : ℕ), A' n ⊆ FiniteSpanningSetsIn.set s n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ n, A' n ** refine'
⟨fun n => A ∩ disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
(inter_subset_right _ _).trans (disjointed_subset _ _),
(disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩ ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} A : Set α hA : MeasurableSet A r : ℝ≥0∞ hr : r > ↑↑μ A hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) ⊢ A = ⋃ n, (fun n => A ∩ disjointed s.set n) n ** rw [← inter_iUnion, iUnion_disjointed, s.spanning, inter_univ] ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r ⊢ ∀ (n : ℕ), ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n ** intro n ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r n : ℕ ⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n ** have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n) ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r n : ℕ H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n) ⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n ** have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
rw [H₁]
exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r n : ℕ H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n) Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤ ⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n ** rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r n : ℕ H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n) Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤ U : Set α hAU : U ⊇ A n hUo : IsOpen U hU : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) U < ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) + δ n ⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n ** rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r n : ℕ H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n) Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤ U : Set α hAU : U ⊇ A n hUo : IsOpen U hU : ↑↑μ (U ∩ FiniteSpanningSetsIn.set s n) < ↑↑μ (A n) + δ n ⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n ** exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩ ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r n : ℕ H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n) ⊢ ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤ ** rw [H₁] ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ✝ : Measure α inst✝ : OpensMeasurableSpace α μ : Measure α s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)} r : ℝ≥0∞ hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n) this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n)) A : ℕ → Set α hAm : ∀ (n : ℕ), MeasurableSet (A n) hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n hAd : Pairwise (Disjoint on A) hA : MeasurableSet (⋃ n, A n) hr : r > ↑↑μ (⋃ n, A n) δ : ℕ → ℝ≥0∞ δ0 : ∀ (i : ℕ), 0 < δ i hδε✝ : ∑' (i : ℕ), δ i < r - ↑↑μ (⋃ n, A n) hδε : ↑↑μ (⋃ n, A n) + ∑' (i : ℕ), δ i < r n : ℕ H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n) ⊢ ↑↑μ (A n ∩ FiniteSpanningSetsIn.set s n) ≠ ⊤ ** exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne ** Qed
| |
MeasureTheory.Measure.InnerRegular.of_pseudoEMetricSpace ** α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α inst✝² : TopologicalSpace α μ✝ : Measure α p q : Set α → Prop U s : Set α ε r : ℝ≥0∞ X : Type u_3 inst✝¹ : PseudoEMetricSpace X inst✝ : MeasurableSpace X μ : Measure X ⊢ InnerRegular μ IsClosed IsOpen ** intro U hU r hr ** α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α inst✝² : TopologicalSpace α μ✝ : Measure α p q : Set α → Prop U✝ s : Set α ε r✝ : ℝ≥0∞ X : Type u_3 inst✝¹ : PseudoEMetricSpace X inst✝ : MeasurableSpace X μ : Measure X U : Set X hU : IsOpen U r : ℝ≥0∞ hr : r < ↑↑μ U ⊢ ∃ K, K ⊆ U ∧ IsClosed K ∧ r < ↑↑μ K ** rcases hU.exists_iUnion_isClosed with ⟨F, F_closed, -, rfl, F_mono⟩ ** case intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α inst✝² : TopologicalSpace α μ✝ : Measure α p q : Set α → Prop U s : Set α ε r✝ : ℝ≥0∞ X : Type u_3 inst✝¹ : PseudoEMetricSpace X inst✝ : MeasurableSpace X μ : Measure X r : ℝ≥0∞ F : ℕ → Set X F_closed : ∀ (n : ℕ), IsClosed (F n) F_mono : Monotone F hU : IsOpen (⋃ n, F n) hr : r < ↑↑μ (⋃ n, F n) ⊢ ∃ K, K ⊆ ⋃ n, F n ∧ IsClosed K ∧ r < ↑↑μ K ** rw [measure_iUnion_eq_iSup F_mono.directed_le] at hr ** case intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α inst✝² : TopologicalSpace α μ✝ : Measure α p q : Set α → Prop U s : Set α ε r✝ : ℝ≥0∞ X : Type u_3 inst✝¹ : PseudoEMetricSpace X inst✝ : MeasurableSpace X μ : Measure X r : ℝ≥0∞ F : ℕ → Set X F_closed : ∀ (n : ℕ), IsClosed (F n) F_mono : Monotone F hU : IsOpen (⋃ n, F n) hr : r < ⨆ i, ↑↑μ (F i) ⊢ ∃ K, K ⊆ ⋃ n, F n ∧ IsClosed K ∧ r < ↑↑μ K ** rcases lt_iSup_iff.1 hr with ⟨n, hn⟩ ** case intro.intro.intro.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α inst✝² : TopologicalSpace α μ✝ : Measure α p q : Set α → Prop U s : Set α ε r✝ : ℝ≥0∞ X : Type u_3 inst✝¹ : PseudoEMetricSpace X inst✝ : MeasurableSpace X μ : Measure X r : ℝ≥0∞ F : ℕ → Set X F_closed : ∀ (n : ℕ), IsClosed (F n) F_mono : Monotone F hU : IsOpen (⋃ n, F n) hr : r < ⨆ i, ↑↑μ (F i) n : ℕ hn : r < ↑↑μ (F n) ⊢ ∃ K, K ⊆ ⋃ n, F n ∧ IsClosed K ∧ r < ↑↑μ K ** exact ⟨F n, subset_iUnion _ _, F_closed n, hn⟩ ** Qed
| |
MeasureTheory.Measure.Regular.exists_compact_not_null ** α : Type u_1 β : Type u_2 inst✝² : MeasurableSpace α inst✝¹ : TopologicalSpace α μ : Measure α inst✝ : Regular μ ⊢ (∃ K, IsCompact K ∧ ↑↑μ K ≠ 0) ↔ μ ≠ 0 ** simp_rw [Ne.def, ← measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact,
ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff] ** Qed
| |
MeasurableSet.exists_isCompact_diff_lt ** α : Type u_1 β : Type u_2 inst✝⁴ : MeasurableSpace α inst✝³ : TopologicalSpace α μ : Measure α inst✝² : OpensMeasurableSpace α inst✝¹ : T2Space α inst✝ : Regular μ A : Set α hA : MeasurableSet A h'A : ↑↑μ A ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ K, K ⊆ A ∧ IsCompact K ∧ ↑↑μ (A \ K) < ε ** rcases hA.exists_isCompact_lt_add h'A hε with ⟨K, hKA, hKc, hK⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝⁴ : MeasurableSpace α inst✝³ : TopologicalSpace α μ : Measure α inst✝² : OpensMeasurableSpace α inst✝¹ : T2Space α inst✝ : Regular μ A : Set α hA : MeasurableSet A h'A : ↑↑μ A ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 K : Set α hKA : K ⊆ A hKc : IsCompact K hK : ↑↑μ A < ↑↑μ K + ε ⊢ ∃ K, K ⊆ A ∧ IsCompact K ∧ ↑↑μ (A \ K) < ε ** exact
⟨K, hKA, hKc,
measure_diff_lt_of_lt_add hKc.measurableSet hKA (ne_top_of_le_ne_top h'A <| measure_mono hKA)
hK⟩ ** Qed
| |
MeasurableSet.exists_isClosed_diff_lt ** α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α inst✝² : TopologicalSpace α μ : Measure α inst✝¹ : OpensMeasurableSpace α inst✝ : WeaklyRegular μ A : Set α hA : MeasurableSet A h'A : ↑↑μ A ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ F, F ⊆ A ∧ IsClosed F ∧ ↑↑μ (A \ F) < ε ** rcases hA.exists_isClosed_lt_add h'A hε with ⟨F, hFA, hFc, hF⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 inst✝³ : MeasurableSpace α inst✝² : TopologicalSpace α μ : Measure α inst✝¹ : OpensMeasurableSpace α inst✝ : WeaklyRegular μ A : Set α hA : MeasurableSet A h'A : ↑↑μ A ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 F : Set α hFA : F ⊆ A hFc : IsClosed F hF : ↑↑μ A < ↑↑μ F + ε ⊢ ∃ F, F ⊆ A ∧ IsClosed F ∧ ↑↑μ (A \ F) < ε ** exact
⟨F, hFA, hFc,
measure_diff_lt_of_lt_add hFc.measurableSet hFA (ne_top_of_le_ne_top h'A <| measure_mono hFA)
hF⟩ ** Qed
| |
MeasureTheory.lintegral_eq_lintegral_meas_le ** α : Type u_1 inst✝ : MeasurableSpace α f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f ⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ** set cst := fun _ : ℝ => (1 : ℝ) ** α : Type u_1 inst✝ : MeasurableSpace α f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f cst : ℝ → ℝ := fun x => 1 ⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ** have cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t := fun _ _ =>
intervalIntegrable_const ** α : Type u_1 inst✝ : MeasurableSpace α f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f cst : ℝ → ℝ := fun x => 1 cst_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable cst volume 0 t ⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ** have key :=
lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble cst_intble
(eventually_of_forall fun _ => zero_le_one) ** α : Type u_1 inst✝ : MeasurableSpace α f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f cst : ℝ → ℝ := fun x => 1 cst_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable cst volume 0 t key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, cst t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (cst t) ⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ** simp_rw [ENNReal.ofReal_one, mul_one] at key ** α : Type u_1 inst✝ : MeasurableSpace α f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f cst : ℝ → ℝ := fun x => 1 cst_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable cst volume 0 t key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ** rw [← key] ** α : Type u_1 inst✝ : MeasurableSpace α f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f cst : ℝ → ℝ := fun x => 1 cst_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable cst volume 0 t key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ ** congr with ω ** case e_f.h α : Type u_1 inst✝ : MeasurableSpace α f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f cst : ℝ → ℝ := fun x => 1 cst_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable cst volume 0 t key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} ω : α ⊢ ENNReal.ofReal (f ω) = ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, 1) ** simp only [intervalIntegral.integral_const, sub_zero, Algebra.id.smul_eq_mul, mul_one] ** Qed
| |
MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul ** α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f g_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t g_nn : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), 0 ≤ g t ⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t < f a} * ENNReal.ofReal (g t) ** rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn] ** α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f g_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t g_nn : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), 0 ≤ g t ⊢ ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t < f a} * ENNReal.ofReal (g t) ** apply lintegral_congr_ae ** case h α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f g_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t g_nn : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), 0 ≤ g t ⊢ (fun a => ↑↑μ {a_1 | a ≤ f a_1} * ENNReal.ofReal (g a)) =ᶠ[ae (Measure.restrict volume (Ioi 0))] fun a => ↑↑μ {a_1 | a < f a_1} * ENNReal.ofReal (g a) ** filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
with t ht ** case h α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f g_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t g_nn : ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioi 0), 0 ≤ g t t : ℝ ht : ↑↑μ {a | t ≤ f a} = ↑↑μ {a | t < f a} ⊢ ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ↑↑μ {a | t < f a} * ENNReal.ofReal (g t) ** rw [ht] ** Qed
| |
MeasureTheory.lintegral_eq_lintegral_meas_lt ** α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f ⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t < f a} ** rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble] ** α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f ⊢ ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} = ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t < f a} ** apply lintegral_congr_ae ** case h α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f ⊢ (fun a => ↑↑μ {a_1 | a ≤ f a_1}) =ᶠ[ae (Measure.restrict volume (Ioi 0))] fun a => ↑↑μ {a_1 | a < f a_1} ** filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
with t ht ** case h α : Type u_1 inst✝² : MeasurableSpace α μ✝ : Measure α β : Type u_2 inst✝¹ : MeasurableSpace β inst✝ : MeasurableSingletonClass β f : α → ℝ g : ℝ → ℝ s : Set α μ : Measure α f_nn : 0 ≤ᶠ[ae μ] f f_mble : AEMeasurable f t : ℝ ht : ↑↑μ {a | t ≤ f a} = ↑↑μ {a | t < f a} ⊢ ↑↑μ {a | t ≤ f a} = ↑↑μ {a | t < f a} ** rw [ht] ** Qed
| |
intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l ⊢ (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ ** have A := hf.integral_sub_linear_isLittleO_ae hfm hl (hu.Ioc hv) ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A✝ : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l A : (fun i => ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) ⊢ (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ ** have B := hf.integral_sub_linear_isLittleO_ae hfm hl (hv.Ioc hu) ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A✝ : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l A : (fun i => ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) B : (fun i => ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) ⊢ (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ ** simp_rw [integral_const', sub_smul] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A✝ : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l A : (fun i => ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) B : (fun i => ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) ⊢ (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - (ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • c - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • c)) =o[lt] fun t => ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • 1 - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • 1 ** refine ((A.trans_le fun t ↦ ?_).sub (B.trans_le fun t ↦ ?_)).congr_left fun t ↦ ?_ ** case refine_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A✝ : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l A : (fun i => ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) B : (fun i => ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) t : ι ⊢ ‖ENNReal.toReal (↑↑μ (Ioc (u t) (v t)))‖ ≤ ‖ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • 1 - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • 1‖ ** cases le_total (u t) (v t) <;> simp [*] ** case refine_2 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A✝ : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l A : (fun i => ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) B : (fun i => ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) t : ι ⊢ ‖ENNReal.toReal (↑↑μ (Ioc (v t) (u t)))‖ ≤ ‖ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • 1 - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • 1‖ ** cases le_total (u t) (v t) <;> simp [*] ** case refine_3 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A✝ : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l A : (fun i => ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) B : (fun i => ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) t : ι ⊢ ∫ (x : ℝ) in Ioc (u t) (v t), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • c - (∫ (x : ℝ) in Ioc (v t) (u t), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • c) = ∫ (x : ℝ) in u t..v t, f x ∂μ - (ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • c - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • c) ** simp_rw [intervalIntegral] ** case refine_3 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A✝ : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l A : (fun i => ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (u i) (v i))) B : (fun i => ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) • c) =o[lt] fun i => ENNReal.toReal (↑↑μ (Ioc (v i) (u i))) t : ι ⊢ ∫ (x : ℝ) in Ioc (u t) (v t), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • c - (∫ (x : ℝ) in Ioc (v t) (u t), f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • c) = ∫ (x : ℝ) in Ioc (u t) (v t), f x ∂μ - ∫ (x : ℝ) in Ioc (v t) (u t), f x ∂μ - (ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • c - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • c) ** abel ** Qed
| |
intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l huv : u ≤ᶠ[lt] v x : ι hx : u x ≤ v x ⊢ (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) x = (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (u t) (v t))) • c) x ** simp [integral_const', hx] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l huv : u ≤ᶠ[lt] v x : ι hx : u x ≤ v x ⊢ (fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ) x = (fun t => ENNReal.toReal (↑↑μ (Ioc (u t) (v t)))) x ** simp [integral_const', hx] ** Qed
| |
intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝¹ : IsMeasurablyGenerated l' inst✝ : TendstoIxxClass Ioc l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae μ) (𝓝 c) hl : Measure.FiniteAtFilter μ l' hu : Tendsto u lt l hv : Tendsto v lt l huv : v ≤ᶠ[lt] u t : ι ⊢ -(∫ (x : ℝ) in v t..u t, f x ∂μ - ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • c) = ∫ (x : ℝ) in u t..v t, f x ∂μ + ENNReal.toReal (↑↑μ (Ioc (v t) (u t))) • c ** simp [integral_symm (u t), add_comm] ** Qed
| |
intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝² : FTCFilter a la la' inst✝¹ : FTCFilter b lb lb' inst✝ : IsLocallyFiniteMeasure μ hab : IntervalIntegrable f μ a b hmeas : StronglyMeasurableAtFilter f lb' hf : Tendsto f (lb' ⊓ Measure.ae μ) (𝓝 c) hu : Tendsto u lt lb hv : Tendsto v lt lb ⊢ (fun t => ∫ (x : ℝ) in a..v t, f x ∂μ - ∫ (x : ℝ) in a..u t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ ** simpa using
measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab stronglyMeasurableAt_bot
hmeas ((tendsto_bot : Tendsto _ ⊥ (𝓝 (0 : E))).mono_left inf_le_left) hf
(tendsto_const_pure : Tendsto _ _ (pure a)) tendsto_const_pure hu hv ** Qed
| |
intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f : ℝ → E a b : ℝ c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι μ : Measure ℝ u v ua va ub vb : ι → ℝ inst✝² : FTCFilter a la la' inst✝¹ : FTCFilter b lb lb' inst✝ : IsLocallyFiniteMeasure μ hab : IntervalIntegrable f μ a b hmeas : StronglyMeasurableAtFilter f la' hf : Tendsto f (la' ⊓ Measure.ae μ) (𝓝 c) hu : Tendsto u lt la hv : Tendsto v lt la ⊢ (fun t => ∫ (x : ℝ) in v t..b, f x ∂μ - ∫ (x : ℝ) in u t..b, f x ∂μ + ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ ** simpa using
measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab hmeas
stronglyMeasurableAt_bot hf ((tendsto_bot : Tendsto _ ⊥ (𝓝 (0 : E))).mono_left inf_le_left) hu
hv (tendsto_const_pure : Tendsto _ _ (pure b)) tendsto_const_pure ** Qed
| |
intervalIntegral.integral_sub_linear_isLittleO_of_tendsto_ae ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u✝ v✝ ua ub va vb : ι → ℝ inst✝² : FTCFilter a la la' inst✝¹ : FTCFilter b lb lb' inst✝ : FTCFilter a l l' hfm : StronglyMeasurableAtFilter f l' hf : Tendsto f (l' ⊓ Measure.ae volume) (𝓝 c) u v : ι → ℝ hu : Tendsto u lt l hv : Tendsto v lt l ⊢ (fun t => (∫ (x : ℝ) in u t..v t, f x) - (v t - u t) • c) =o[lt] (v - u) ** simpa [integral_const] using measure_integral_sub_linear_isLittleO_of_tendsto_ae hfm hf hu hv ** Qed
| |
intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hab : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f la' hmeas_b : StronglyMeasurableAtFilter f lb' ha_lim : Tendsto f (la' ⊓ Measure.ae volume) (𝓝 ca) hb_lim : Tendsto f (lb' ⊓ Measure.ae volume) (𝓝 cb) hua : Tendsto ua lt la hva : Tendsto va lt la hub : Tendsto ub lt lb hvb : Tendsto vb lt lb ⊢ (fun t => ((∫ (x : ℝ) in va t..vb t, f x) - ∫ (x : ℝ) in ua t..ub t, f x) - ((vb t - ub t) • cb - (va t - ua t) • ca)) =o[lt] fun t => ‖va t - ua t‖ + ‖vb t - ub t‖ ** simpa [integral_const]
using measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hab hmeas_a hmeas_b
ha_lim hb_lim hua hva hub hvb ** Qed
| |
intervalIntegral.integral_hasStrictFDerivAt_of_tendsto_ae ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) ha : Tendsto f (𝓝 a ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (𝓝 b ⊓ Measure.ae volume) (𝓝 cb) ⊢ HasStrictFDerivAt (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (a, b) ** have :=
integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hf hmeas_a hmeas_b ha hb
(continuous_snd.fst.tendsto ((a, b), (a, b)))
(continuous_fst.fst.tendsto ((a, b), (a, b)))
(continuous_snd.snd.tendsto ((a, b), (a, b)))
(continuous_fst.snd.tendsto ((a, b), (a, b))) ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) ha : Tendsto f (𝓝 a ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (𝓝 b ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1.1 ..t.1.2, f x) - ∫ (x : ℝ) in t.2.1 ..t.2.2, f x) - ((t.1.2 - t.2.2) • cb - (t.1.1 - t.2.1) • ca)) =o[𝓝 ((a, b), a, b)] fun t => ‖t.1.1 - t.2.1‖ + ‖t.1.2 - t.2.2‖ ⊢ HasStrictFDerivAt (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (a, b) ** refine' (this.congr_left _).trans_isBigO _ ** case refine'_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) ha : Tendsto f (𝓝 a ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (𝓝 b ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1.1 ..t.1.2, f x) - ∫ (x : ℝ) in t.2.1 ..t.2.2, f x) - ((t.1.2 - t.2.2) • cb - (t.1.1 - t.2.1) • ca)) =o[𝓝 ((a, b), a, b)] fun t => ‖t.1.1 - t.2.1‖ + ‖t.1.2 - t.2.2‖ ⊢ ∀ (x : (ℝ × ℝ) × ℝ × ℝ), ((∫ (x : ℝ) in x.1.1 ..x.1.2, f x) - ∫ (x : ℝ) in x.2.1 ..x.2.2, f x) - ((x.1.2 - x.2.2) • cb - (x.1.1 - x.2.1) • ca) = (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) x.1 - (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) x.2 - ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x.1 - x.2) ** intro x ** case refine'_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) ha : Tendsto f (𝓝 a ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (𝓝 b ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1.1 ..t.1.2, f x) - ∫ (x : ℝ) in t.2.1 ..t.2.2, f x) - ((t.1.2 - t.2.2) • cb - (t.1.1 - t.2.1) • ca)) =o[𝓝 ((a, b), a, b)] fun t => ‖t.1.1 - t.2.1‖ + ‖t.1.2 - t.2.2‖ x : (ℝ × ℝ) × ℝ × ℝ ⊢ ((∫ (x : ℝ) in x.1.1 ..x.1.2, f x) - ∫ (x : ℝ) in x.2.1 ..x.2.2, f x) - ((x.1.2 - x.2.2) • cb - (x.1.1 - x.2.1) • ca) = (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) x.1 - (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) x.2 - ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x.1 - x.2) ** simp [sub_smul] ** case refine'_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) ha : Tendsto f (𝓝 a ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (𝓝 b ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1.1 ..t.1.2, f x) - ∫ (x : ℝ) in t.2.1 ..t.2.2, f x) - ((t.1.2 - t.2.2) • cb - (t.1.1 - t.2.1) • ca)) =o[𝓝 ((a, b), a, b)] fun t => ‖t.1.1 - t.2.1‖ + ‖t.1.2 - t.2.2‖ x : (ℝ × ℝ) × ℝ × ℝ ⊢ x.1.2 • cb - x.2.2 • cb - (x.1.1 • ca - x.2.1 • ca) = x.1.2 • cb - x.1.1 • ca - (x.2.2 • cb - x.2.1 • ca) ** abel ** case refine'_2 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) ha : Tendsto f (𝓝 a ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (𝓝 b ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1.1 ..t.1.2, f x) - ∫ (x : ℝ) in t.2.1 ..t.2.2, f x) - ((t.1.2 - t.2.2) • cb - (t.1.1 - t.2.1) • ca)) =o[𝓝 ((a, b), a, b)] fun t => ‖t.1.1 - t.2.1‖ + ‖t.1.2 - t.2.2‖ ⊢ (fun t => ‖t.1.1 - t.2.1‖ + ‖t.1.2 - t.2.2‖) =O[𝓝 ((a, b), a, b)] fun p => p.1 - p.2 ** exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left ** Qed
| |
intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_left ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas : StronglyMeasurableAtFilter f (𝓝 a) ha : Tendsto f (𝓝 a ⊓ Measure.ae volume) (𝓝 c) ⊢ HasStrictDerivAt (fun u => ∫ (x : ℝ) in u..b, f x) (-c) a ** simpa only [← integral_symm] using
(integral_hasStrictDerivAt_of_tendsto_ae_right hf.symm hmeas ha).neg ** Qed
| |
intervalIntegral.integral_hasStrictDerivAt_left ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : CompleteSpace E inst✝² : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝¹ : FTCFilter a la la' inst✝ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b hmeas : StronglyMeasurableAtFilter f (𝓝 a) ha : ContinuousAt f a ⊢ HasStrictDerivAt (fun u => ∫ (x : ℝ) in u..b, f x) (-f a) a ** simpa only [← integral_symm] using (integral_hasStrictDerivAt_right hf.symm hmeas ha).neg ** Qed
| |
intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝³ : FTCFilter a la la' inst✝² : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝¹ : FTCFilter a (𝓝[s] a) la inst✝ : FTCFilter b (𝓝[t] b) lb hmeas_a : StronglyMeasurableAtFilter f la hmeas_b : StronglyMeasurableAtFilter f lb ha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb) ⊢ HasFDerivWithinAt (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (s ×ˢ t) (a, b) ** rw [HasFDerivWithinAt, nhdsWithin_prod_eq] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝³ : FTCFilter a la la' inst✝² : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝¹ : FTCFilter a (𝓝[s] a) la inst✝ : FTCFilter b (𝓝[t] b) lb hmeas_a : StronglyMeasurableAtFilter f la hmeas_b : StronglyMeasurableAtFilter f lb ha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb) ⊢ HasFDerivAtFilter (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (a, b) (𝓝[s] a ×ˢ 𝓝[t] b) ** have :=
integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hf hmeas_a hmeas_b ha hb
(tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[s] a)) tendsto_fst
(tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[t] b)) tendsto_snd ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝³ : FTCFilter a la la' inst✝² : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝¹ : FTCFilter a (𝓝[s] a) la inst✝ : FTCFilter b (𝓝[t] b) lb hmeas_a : StronglyMeasurableAtFilter f la hmeas_b : StronglyMeasurableAtFilter f lb ha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1 ..t.2, f x) - ∫ (x : ℝ) in a..b, f x) - ((t.2 - b) • cb - (t.1 - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b] fun t => ‖t.1 - a‖ + ‖t.2 - b‖ ⊢ HasFDerivAtFilter (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (a, b) (𝓝[s] a ×ˢ 𝓝[t] b) ** refine' (this.congr_left _).trans_isBigO _ ** case refine'_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝³ : FTCFilter a la la' inst✝² : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝¹ : FTCFilter a (𝓝[s] a) la inst✝ : FTCFilter b (𝓝[t] b) lb hmeas_a : StronglyMeasurableAtFilter f la hmeas_b : StronglyMeasurableAtFilter f lb ha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1 ..t.2, f x) - ∫ (x : ℝ) in a..b, f x) - ((t.2 - b) • cb - (t.1 - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b] fun t => ‖t.1 - a‖ + ‖t.2 - b‖ ⊢ ∀ (x : ℝ × ℝ), ((∫ (x : ℝ) in x.1 ..x.2, f x) - ∫ (x : ℝ) in a..b, f x) - ((x.2 - b) • cb - (x.1 - a) • ca) = (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) x - (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (a, b) - ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x - (a, b)) ** intro x ** case refine'_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝³ : FTCFilter a la la' inst✝² : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝¹ : FTCFilter a (𝓝[s] a) la inst✝ : FTCFilter b (𝓝[t] b) lb hmeas_a : StronglyMeasurableAtFilter f la hmeas_b : StronglyMeasurableAtFilter f lb ha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1 ..t.2, f x) - ∫ (x : ℝ) in a..b, f x) - ((t.2 - b) • cb - (t.1 - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b] fun t => ‖t.1 - a‖ + ‖t.2 - b‖ x : ℝ × ℝ ⊢ ((∫ (x : ℝ) in x.1 ..x.2, f x) - ∫ (x : ℝ) in a..b, f x) - ((x.2 - b) • cb - (x.1 - a) • ca) = (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) x - (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (a, b) - ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x - (a, b)) ** simp [sub_smul] ** case refine'_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝³ : FTCFilter a la la' inst✝² : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝¹ : FTCFilter a (𝓝[s] a) la inst✝ : FTCFilter b (𝓝[t] b) lb hmeas_a : StronglyMeasurableAtFilter f la hmeas_b : StronglyMeasurableAtFilter f lb ha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1 ..t.2, f x) - ∫ (x : ℝ) in a..b, f x) - ((t.2 - b) • cb - (t.1 - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b] fun t => ‖t.1 - a‖ + ‖t.2 - b‖ x : ℝ × ℝ ⊢ x.2 • cb - b • cb - (x.1 • ca - a • ca) = x.2 • cb - x.1 • ca - (b • cb - a • ca) ** abel ** case refine'_2 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : CompleteSpace E inst✝⁴ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝³ : FTCFilter a la la' inst✝² : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝¹ : FTCFilter a (𝓝[s] a) la inst✝ : FTCFilter b (𝓝[t] b) lb hmeas_a : StronglyMeasurableAtFilter f la hmeas_b : StronglyMeasurableAtFilter f lb ha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca) hb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb) this : (fun t => ((∫ (x : ℝ) in t.1 ..t.2, f x) - ∫ (x : ℝ) in a..b, f x) - ((t.2 - b) • cb - (t.1 - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b] fun t => ‖t.1 - a‖ + ‖t.2 - b‖ ⊢ (fun t => ‖t.1 - a‖ + ‖t.2 - b‖) =O[𝓝[s] a ×ˢ 𝓝[t] b] fun x' => x' - (a, b) ** exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left ** Qed
| |
intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_left ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝² : FTCFilter a la la' inst✝¹ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝ : FTCFilter a (𝓝[s] a) (𝓝[t] a) hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) ha : Tendsto f (𝓝[t] a ⊓ Measure.ae volume) (𝓝 c) ⊢ HasDerivWithinAt (fun u => ∫ (x : ℝ) in u..b, f x) (-c) s a ** simp only [integral_symm b] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f : ℝ → E c ca cb : E l l' la la' lb lb' : Filter ℝ lt : Filter ι a b z : ℝ u v ua ub va vb : ι → ℝ inst✝² : FTCFilter a la la' inst✝¹ : FTCFilter b lb lb' hf : IntervalIntegrable f volume a b s t : Set ℝ inst✝ : FTCFilter a (𝓝[s] a) (𝓝[t] a) hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) ha : Tendsto f (𝓝[t] a ⊓ Measure.ae volume) (𝓝 c) ⊢ HasDerivWithinAt (fun u => -∫ (x : ℝ) in b..u, f x) (-c) s a ** exact (integral_hasDerivWithinAt_of_tendsto_ae_right hf.symm hmeas ha).neg ** Qed
| |
intervalIntegral.integral_le_sub_of_hasDeriv_right_of_le ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : ℝ → E g' g φ : ℝ → ℝ a b : ℝ hab : a ≤ b hcont : ContinuousOn g (Icc a b) hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x φint : IntegrableOn φ (Icc a b) hφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x ⊢ ∫ (y : ℝ) in a..b, φ y ≤ g b - g a ** rw [← neg_le_neg_iff] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : ℝ → E g' g φ : ℝ → ℝ a b : ℝ hab : a ≤ b hcont : ContinuousOn g (Icc a b) hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x φint : IntegrableOn φ (Icc a b) hφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x ⊢ -(g b - g a) ≤ -∫ (y : ℝ) in a..b, φ y ** convert sub_le_integral_of_hasDeriv_right_of_le hab hcont.neg (fun x hx => (hderiv x hx).neg)
φint.neg fun x hx => neg_le_neg (hφg x hx) using 1 ** case h.e'_3 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : ℝ → E g' g φ : ℝ → ℝ a b : ℝ hab : a ≤ b hcont : ContinuousOn g (Icc a b) hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x φint : IntegrableOn φ (Icc a b) hφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x ⊢ -(g b - g a) = -g b - -g a ** abel ** case h.e'_4 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : ℝ → E g' g φ : ℝ → ℝ a b : ℝ hab : a ≤ b hcont : ContinuousOn g (Icc a b) hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x φint : IntegrableOn φ (Icc a b) hφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x ⊢ -∫ (y : ℝ) in a..b, φ y = ∫ (y : ℝ) in a..b, (-φ) y ** simp only [← integral_neg] ** case h.e'_4 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : ℝ → E g' g φ : ℝ → ℝ a b : ℝ hab : a ≤ b hcont : ContinuousOn g (Icc a b) hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt g (g' x) (Ioi x) x φint : IntegrableOn φ (Icc a b) hφg : ∀ (x : ℝ), x ∈ Ioo a b → φ x ≤ g' x ⊢ ∫ (x : ℝ) in a..b, -φ x = ∫ (y : ℝ) in a..b, (-φ) y ** rfl ** Qed
| |
intervalIntegral.integral_eq_sub_of_hasDeriv_right ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hcont : ContinuousOn f [[a, b]] hderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hint : IntervalIntegrable f' volume a b ⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a ** cases' le_total a b with hab hab ** case inl ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hcont : ContinuousOn f [[a, b]] hderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hint : IntervalIntegrable f' volume a b hab : a ≤ b ⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a ** simp only [uIcc_of_le, min_eq_left, max_eq_right, hab] at hcont hderiv hint ** case inl ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hint : IntervalIntegrable f' volume a b hab : a ≤ b hcont : ContinuousOn f (Icc a b) hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt f (f' x) (Ioi x) x ⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a ** apply integral_eq_sub_of_hasDeriv_right_of_le hab hcont hderiv hint ** case inr ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hcont : ContinuousOn f [[a, b]] hderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hint : IntervalIntegrable f' volume a b hab : b ≤ a ⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a ** simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab] at hcont hderiv ** case inr ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hint : IntervalIntegrable f' volume a b hab : b ≤ a hcont : ContinuousOn f (Icc b a) hderiv : ∀ (x : ℝ), x ∈ Ioo b a → HasDerivWithinAt f (f' x) (Ioi x) x ⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a ** rw [integral_symm, integral_eq_sub_of_hasDeriv_right_of_le hab hcont hderiv hint.symm, neg_sub] ** Qed
| |
intervalIntegral.integral_eq_sub_of_hasDerivAt_of_tendsto ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) ⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa ** set F : ℝ → E := update (update f a fa) b fb ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb ⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa ** have Fderiv : ∀ x ∈ Ioo a b, HasDerivAt F (f' x) x := by
refine' fun x hx => (hderiv x hx).congr_of_eventuallyEq _
filter_upwards [Ioo_mem_nhds hx.1 hx.2] with _ hy
unfold_let F
rw [update_noteq hy.2.ne, update_noteq hy.1.ne'] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb Fderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x hcont : ContinuousOn F (Icc a b) ⊢ ∫ (y : ℝ) in a..b, f' y = fb - fa ** simpa [hab.ne, hab.ne'] using integral_eq_sub_of_hasDerivAt_of_le hab.le hcont Fderiv hint ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb ⊢ ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x ** refine' fun x hx => (hderiv x hx).congr_of_eventuallyEq _ ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb x : ℝ hx : x ∈ Ioo a b ⊢ F =ᶠ[𝓝 x] f ** filter_upwards [Ioo_mem_nhds hx.1 hx.2] with _ hy ** case h ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb x : ℝ hx : x ∈ Ioo a b a✝ : ℝ hy : a✝ ∈ Ioo a b ⊢ F a✝ = f a✝ ** unfold_let F ** case h ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb x : ℝ hx : x ∈ Ioo a b a✝ : ℝ hy : a✝ ∈ Ioo a b ⊢ update (update f a fa) b fb a✝ = f a✝ ** rw [update_noteq hy.2.ne, update_noteq hy.1.ne'] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb Fderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x ⊢ ContinuousOn F (Icc a b) ** rw [continuousOn_update_iff, continuousOn_update_iff, Icc_diff_right, Ico_diff_left] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb Fderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x ⊢ (ContinuousOn f (Ioo a b) ∧ (a ∈ Ico a b → Tendsto f (𝓝[Ioo a b] a) (𝓝 fa))) ∧ (b ∈ Icc a b → Tendsto (update f a fa) (𝓝[Ico a b] b) (𝓝 fb)) ** refine' ⟨⟨fun z hz => (hderiv z hz).continuousAt.continuousWithinAt, _⟩, _⟩ ** case refine'_1 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb Fderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x ⊢ a ∈ Ico a b → Tendsto f (𝓝[Ioo a b] a) (𝓝 fa) ** exact fun _ => ha.mono_left (nhdsWithin_mono _ Ioo_subset_Ioi_self) ** case refine'_2 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb Fderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x ⊢ b ∈ Icc a b → Tendsto (update f a fa) (𝓝[Ico a b] b) (𝓝 fb) ** rintro - ** case refine'_2 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb Fderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x ⊢ Tendsto (update f a fa) (𝓝[Ico a b] b) (𝓝 fb) ** refine' (hb.congr' _).mono_left (nhdsWithin_mono _ Ico_subset_Iio_self) ** case refine'_2 ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F✝ : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a < b fa fb : E hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x hint : IntervalIntegrable f' volume a b ha : Tendsto f (𝓝[Ioi a] a) (𝓝 fa) hb : Tendsto f (𝓝[Iio b] b) (𝓝 fb) F : ℝ → E := update (update f a fa) b fb Fderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt F (f' x) x ⊢ f =ᶠ[𝓝[Iio b] b] update f a fa ** filter_upwards [Ioo_mem_nhdsWithin_Iio (right_mem_Ioc.2 hab)] with _ hz using
(update_noteq hz.1.ne' _ _).symm ** Qed
| |
intervalIntegral.integral_deriv_eq_sub' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g φ : ℝ → ℝ f✝ f' : ℝ → E a b : ℝ f : ℝ → E hderiv : deriv f = f' hdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x hcont : ContinuousOn f' [[a, b]] ⊢ ∫ (y : ℝ) in a..b, f' y = f b - f a ** rw [← hderiv, integral_deriv_eq_sub hdiff] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g φ : ℝ → ℝ f✝ f' : ℝ → E a b : ℝ f : ℝ → E hderiv : deriv f = f' hdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x hcont : ContinuousOn f' [[a, b]] ⊢ IntervalIntegrable (deriv f) volume a b ** rw [hderiv] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g φ : ℝ → ℝ f✝ f' : ℝ → E a b : ℝ f : ℝ → E hderiv : deriv f = f' hdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x hcont : ContinuousOn f' [[a, b]] ⊢ IntervalIntegrable f' volume a b ** exact hcont.intervalIntegrable ** Qed
| |
intervalIntegral.intervalIntegrable_deriv_of_nonneg ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hcont : ContinuousOn g [[a, b]] hderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivAt g (g' x) x hpos : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → 0 ≤ g' x ⊢ IntervalIntegrable g' volume a b ** cases' le_total a b with hab hab ** case inl ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hcont : ContinuousOn g [[a, b]] hderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivAt g (g' x) x hpos : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → 0 ≤ g' x hab : a ≤ b ⊢ IntervalIntegrable g' volume a b ** simp only [uIcc_of_le, min_eq_left, max_eq_right, hab, IntervalIntegrable, hab,
Ioc_eq_empty_of_le, integrableOn_empty, and_true_iff] at hcont hderiv hpos ⊢ ** case inl ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : a ≤ b hcont : ContinuousOn g (Icc a b) hderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x hpos : ∀ (x : ℝ), x ∈ Ioo a b → 0 ≤ g' x ⊢ IntegrableOn g' (Ioc a b) ** exact integrableOn_deriv_of_nonneg hcont hderiv hpos ** case inr ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hcont : ContinuousOn g [[a, b]] hderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivAt g (g' x) x hpos : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → 0 ≤ g' x hab : b ≤ a ⊢ IntervalIntegrable g' volume a b ** simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, IntervalIntegrable, Ioc_eq_empty_of_le,
integrableOn_empty, true_and_iff] at hcont hderiv hpos ⊢ ** case inr ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E a b : ℝ hab : b ≤ a hcont : ContinuousOn g (Icc b a) hderiv : ∀ (x : ℝ), x ∈ Ioo b a → HasDerivAt g (g' x) x hpos : ∀ (x : ℝ), x ∈ Ioo b a → 0 ≤ g' x ⊢ IntegrableOn g' (Ioc b a) ** exact integrableOn_deriv_of_nonneg hcont hderiv hpos ** Qed
| |
intervalIntegral.integral_mul_deriv_eq_deriv_mul ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E inst✝² : NormedRing A inst✝¹ : NormedAlgebra ℝ A inst✝ : CompleteSpace A a b : ℝ u v u' v' : ℝ → A hu : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt u (u' x) x hv : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt v (v' x) x hu' : IntervalIntegrable u' volume a b hv' : IntervalIntegrable v' volume a b ⊢ ∫ (x : ℝ) in a..b, u x * v' x = u b * v b - u a * v a - ∫ (x : ℝ) in a..b, u' x * v x ** rw [← integral_deriv_mul_eq_sub hu hv hu' hv', ← integral_sub] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E inst✝² : NormedRing A inst✝¹ : NormedAlgebra ℝ A inst✝ : CompleteSpace A a b : ℝ u v u' v' : ℝ → A hu : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt u (u' x) x hv : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt v (v' x) x hu' : IntervalIntegrable u' volume a b hv' : IntervalIntegrable v' volume a b ⊢ ∫ (x : ℝ) in a..b, u x * v' x = ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x - u' x * v x ** exact integral_congr fun x _ => by simp only [add_sub_cancel'] ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E inst✝² : NormedRing A inst✝¹ : NormedAlgebra ℝ A inst✝ : CompleteSpace A a b : ℝ u v u' v' : ℝ → A hu : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt u (u' x) x hv : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt v (v' x) x hu' : IntervalIntegrable u' volume a b hv' : IntervalIntegrable v' volume a b x : ℝ x✝ : x ∈ [[a, b]] ⊢ u x * v' x = u' x * v x + u x * v' x - u' x * v x ** simp only [add_sub_cancel'] ** case hf ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E inst✝² : NormedRing A inst✝¹ : NormedAlgebra ℝ A inst✝ : CompleteSpace A a b : ℝ u v u' v' : ℝ → A hu : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt u (u' x) x hv : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt v (v' x) x hu' : IntervalIntegrable u' volume a b hv' : IntervalIntegrable v' volume a b ⊢ IntervalIntegrable (fun x => u' x * v x + u x * v' x) volume a b ** exact
(hu'.mul_continuousOn (HasDerivAt.continuousOn hv)).add
(hv'.continuousOn_mul (HasDerivAt.continuousOn hu)) ** case hg ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : CompleteSpace E inst✝³ : NormedSpace ℝ E f✝ : ℝ → E g' g φ : ℝ → ℝ f f' : ℝ → E inst✝² : NormedRing A inst✝¹ : NormedAlgebra ℝ A inst✝ : CompleteSpace A a b : ℝ u v u' v' : ℝ → A hu : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt u (u' x) x hv : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt v (v' x) x hu' : IntervalIntegrable u' volume a b hv' : IntervalIntegrable v' volume a b ⊢ IntervalIntegrable (fun x => u' x * v x) volume a b ** exact hu'.mul_continuousOn (HasDerivAt.continuousOn hv) ** Qed
| |
intervalIntegral.integral_comp_smul_deriv'' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' : ℝ → ℝ g : ℝ → E hf : ContinuousOn f [[a, b]] hff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hf' : ContinuousOn f' [[a, b]] hg : ContinuousOn g (f '' [[a, b]]) ⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u ** refine'
integral_comp_smul_deriv''' hf hff' (hg.mono <| image_subset _ Ioo_subset_Icc_self) _
(hf'.smul (hg.comp hf <| subset_preimage_image f _)).integrableOn_Icc ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' : ℝ → ℝ g : ℝ → E hf : ContinuousOn f [[a, b]] hff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hf' : ContinuousOn f' [[a, b]] hg : ContinuousOn g (f '' [[a, b]]) ⊢ IntegrableOn g (f '' [[a, b]]) ** rw [hf.image_uIcc] at hg ⊢ ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' : ℝ → ℝ g : ℝ → E hf : ContinuousOn f [[a, b]] hff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hf' : ContinuousOn f' [[a, b]] hg : ContinuousOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] ⊢ IntegrableOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] ** exact hg.integrableOn_Icc ** Qed
| |
intervalIntegral.integral_comp_mul_deriv''' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' g : ℝ → ℝ hf : ContinuousOn f [[a, b]] hff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntegrableOn g (f '' [[a, b]]) hg2 : IntegrableOn (fun x => (g ∘ f) x * f' x) [[a, b]] ⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u ** have hg2' : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] := by simpa [mul_comm] using hg2 ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' g : ℝ → ℝ hf : ContinuousOn f [[a, b]] hff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntegrableOn g (f '' [[a, b]]) hg2 : IntegrableOn (fun x => (g ∘ f) x * f' x) [[a, b]] hg2' : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] ⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u ** simpa [mul_comm] using integral_comp_smul_deriv''' hf hff' hg_cont hg1 hg2' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' g : ℝ → ℝ hf : ContinuousOn f [[a, b]] hff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b)) hg1 : IntegrableOn g (f '' [[a, b]]) hg2 : IntegrableOn (fun x => (g ∘ f) x * f' x) [[a, b]] ⊢ IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] ** simpa [mul_comm] using hg2 ** Qed
| |
intervalIntegral.integral_comp_mul_deriv'' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' g : ℝ → ℝ hf : ContinuousOn f [[a, b]] hff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x hf' : ContinuousOn f' [[a, b]] hg : ContinuousOn g (f '' [[a, b]]) ⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u ** simpa [mul_comm] using integral_comp_smul_deriv'' hf hff' hf' hg ** Qed
| |
intervalIntegral.integral_comp_mul_deriv' ** ι : Type u_1 𝕜 : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst✝² : NormedAddCommGroup E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f✝¹ : ℝ → E g' g✝ φ : ℝ → ℝ f✝ f'✝ : ℝ → E a b : ℝ f f' g : ℝ → ℝ h : ∀ (x : ℝ), x ∈ [[a, b]] → HasDerivAt f (f' x) x h' : ContinuousOn f' [[a, b]] hg : ContinuousOn g (f '' [[a, b]]) ⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x ** simpa [mul_comm] using integral_comp_smul_deriv' h h' hg ** Qed
| |
MeasureTheory.weightedSMul_apply ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α s : Set α x : F ⊢ ↑(weightedSMul μ s) x = ENNReal.toReal (↑↑μ s) • x ** simp [weightedSMul] ** Qed
| |
MeasureTheory.weightedSMul_zero_measure ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ : Measure α m : MeasurableSpace α ⊢ weightedSMul 0 = 0 ** ext1 ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ : Measure α m : MeasurableSpace α x✝ : Set α ⊢ weightedSMul 0 x✝ = OfNat.ofNat 0 x✝ ** simp [weightedSMul] ** Qed
| |
MeasureTheory.weightedSMul_empty ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α ⊢ weightedSMul μ ∅ = 0 ** ext1 x ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α x : F ⊢ ↑(weightedSMul μ ∅) x = ↑0 x ** rw [weightedSMul_apply] ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α x : F ⊢ ENNReal.toReal (↑↑μ ∅) • x = ↑0 x ** simp ** Qed
| |
MeasureTheory.weightedSMul_add_measure ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ ν : Measure α s : Set α hμs : ↑↑μ s ≠ ⊤ hνs : ↑↑ν s ≠ ⊤ ⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s ** ext1 x ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ ν : Measure α s : Set α hμs : ↑↑μ s ≠ ⊤ hνs : ↑↑ν s ≠ ⊤ x : F ⊢ ↑(weightedSMul (μ + ν) s) x = ↑(weightedSMul μ s + weightedSMul ν s) x ** push_cast ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ ν : Measure α s : Set α hμs : ↑↑μ s ≠ ⊤ hνs : ↑↑ν s ≠ ⊤ x : F ⊢ ↑(weightedSMul (μ + ν) s) x = (↑(weightedSMul μ s) + ↑(weightedSMul ν s)) x ** simp_rw [Pi.add_apply, weightedSMul_apply] ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ ν : Measure α s : Set α hμs : ↑↑μ s ≠ ⊤ hνs : ↑↑ν s ≠ ⊤ x : F ⊢ ENNReal.toReal (↑↑(μ + ν) s) • x = ENNReal.toReal (↑↑μ s) • x + ENNReal.toReal (↑↑ν s) • x ** push_cast ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ ν : Measure α s : Set α hμs : ↑↑μ s ≠ ⊤ hνs : ↑↑ν s ≠ ⊤ x : F ⊢ ENNReal.toReal ((↑↑μ + ↑↑ν) s) • x = ENNReal.toReal (↑↑μ s) • x + ENNReal.toReal (↑↑ν s) • x ** rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul] ** Qed
| |
MeasureTheory.weightedSMul_smul_measure ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α c : ℝ≥0∞ s : Set α ⊢ weightedSMul (c • μ) s = ENNReal.toReal c • weightedSMul μ s ** ext1 x ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α c : ℝ≥0∞ s : Set α x : F ⊢ ↑(weightedSMul (c • μ) s) x = ↑(ENNReal.toReal c • weightedSMul μ s) x ** push_cast ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α c : ℝ≥0∞ s : Set α x : F ⊢ ↑(weightedSMul (c • μ) s) x = (ENNReal.toReal c • ↑(weightedSMul μ s)) x ** simp_rw [Pi.smul_apply, weightedSMul_apply] ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α c : ℝ≥0∞ s : Set α x : F ⊢ ENNReal.toReal (↑↑(c • μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x ** push_cast ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α c : ℝ≥0∞ s : Set α x : F ⊢ ENNReal.toReal ((c • ↑↑μ) s) • x = ENNReal.toReal c • ENNReal.toReal (↑↑μ s) • x ** simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul] ** Qed
| |
MeasureTheory.weightedSMul_congr ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s t : Set α hst : ↑↑μ s = ↑↑μ t ⊢ weightedSMul μ s = weightedSMul μ t ** ext1 x ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s t : Set α hst : ↑↑μ s = ↑↑μ t x : F ⊢ ↑(weightedSMul μ s) x = ↑(weightedSMul μ t) x ** simp_rw [weightedSMul_apply] ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s t : Set α hst : ↑↑μ s = ↑↑μ t x : F ⊢ ENNReal.toReal (↑↑μ s) • x = ENNReal.toReal (↑↑μ t) • x ** congr 2 ** Qed
| |
MeasureTheory.weightedSMul_null ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s : Set α h_zero : ↑↑μ s = 0 ⊢ weightedSMul μ s = 0 ** ext1 x ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s : Set α h_zero : ↑↑μ s = 0 x : F ⊢ ↑(weightedSMul μ s) x = ↑0 x ** rw [weightedSMul_apply, h_zero] ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s : Set α h_zero : ↑↑μ s = 0 x : F ⊢ ENNReal.toReal 0 • x = ↑0 x ** simp ** Qed
| |
MeasureTheory.weightedSMul_union' ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s t : Set α ht : MeasurableSet t hs_finite : ↑↑μ s ≠ ⊤ ht_finite : ↑↑μ t ≠ ⊤ h_inter : s ∩ t = ∅ ⊢ weightedSMul μ (s ∪ t) = weightedSMul μ s + weightedSMul μ t ** ext1 x ** case h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s t : Set α ht : MeasurableSet t hs_finite : ↑↑μ s ≠ ⊤ ht_finite : ↑↑μ t ≠ ⊤ h_inter : s ∩ t = ∅ x : F ⊢ ↑(weightedSMul μ (s ∪ t)) x = ↑(weightedSMul μ s + weightedSMul μ t) x ** simp_rw [add_apply, weightedSMul_apply,
measure_union (Set.disjoint_iff_inter_eq_empty.mpr h_inter) ht,
ENNReal.toReal_add hs_finite ht_finite, add_smul] ** Qed
| |
MeasureTheory.weightedSMul_smul ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α inst✝² : NormedField 𝕜 inst✝¹ : NormedSpace 𝕜 F inst✝ : SMulCommClass ℝ 𝕜 F c : 𝕜 s : Set α x : F ⊢ ↑(weightedSMul μ s) (c • x) = c • ↑(weightedSMul μ s) x ** simp_rw [weightedSMul_apply, smul_comm] ** Qed
| |
MeasureTheory.weightedSMul_nonneg ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s : Set α x : ℝ hx : 0 ≤ x ⊢ 0 ≤ ↑(weightedSMul μ s) x ** simp only [weightedSMul, Algebra.id.smul_eq_mul, coe_smul', id.def, coe_id', Pi.smul_apply] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F m : MeasurableSpace α μ : Measure α s : Set α x : ℝ hx : 0 ≤ x ⊢ 0 ≤ ENNReal.toReal (↑↑μ s) * x ** exact mul_nonneg toReal_nonneg hx ** Qed
| |
MeasureTheory.SimpleFunc.posPart_map_norm ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ ⊢ map norm (posPart f) = posPart f ** ext ** case H α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ a✝ : α ⊢ ↑(map norm (posPart f)) a✝ = ↑(posPart f) a✝ ** rw [map_apply, Real.norm_eq_abs, abs_of_nonneg] ** case H α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ a✝ : α ⊢ 0 ≤ ↑(posPart f) a✝ ** exact le_max_right _ _ ** Qed
| |
MeasureTheory.SimpleFunc.negPart_map_norm ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ ⊢ map norm (negPart f) = negPart f ** rw [negPart] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ ⊢ map norm (posPart (-f)) = posPart (-f) ** exact posPart_map_norm _ ** Qed
| |
MeasureTheory.SimpleFunc.posPart_sub_negPart ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ ⊢ posPart f - negPart f = f ** simp only [posPart, negPart] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ ⊢ map (fun b => max b 0) f - map (fun b => max b 0) (-f) = f ** ext a ** case H α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ a : α ⊢ ↑(map (fun b => max b 0) f - map (fun b => max b 0) (-f)) a = ↑f a ** rw [coe_sub] ** case H α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝² : LinearOrder E inst✝¹ : Zero E inst✝ : MeasurableSpace α f : α →ₛ ℝ a : α ⊢ (↑(map (fun b => max b 0) f) - ↑(map (fun b => max b 0) (-f))) a = ↑f a ** exact max_zero_sub_eq_self (f a) ** Qed
| |
MeasureTheory.SimpleFunc.integral_eq ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α μ : Measure α f : α →ₛ F ⊢ integral μ f = ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x ** simp [integral, setToSimpleFunc, weightedSMul_apply] ** Qed
| |
MeasureTheory.SimpleFunc.integral_eq_sum_filter ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 m : MeasurableSpace α f : α →ₛ F μ : Measure α ⊢ integral μ f = ∑ x in filter (fun x => x ≠ 0) (SimpleFunc.range f), ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x ** rw [integral_def, setToSimpleFunc_eq_sum_filter] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 m : MeasurableSpace α f : α →ₛ F μ : Measure α ⊢ ∑ x in filter (fun x => x ≠ 0) (SimpleFunc.range f), ↑(weightedSMul μ (↑f ⁻¹' {x})) x = ∑ x in filter (fun x => x ≠ 0) (SimpleFunc.range f), ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x ** simp_rw [weightedSMul_apply] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 m : MeasurableSpace α f : α →ₛ F μ : Measure α ⊢ ∑ x in filter (fun x => x ≠ 0) (SimpleFunc.range f), ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x = ∑ x in filter (fun x => x ≠ 0) (SimpleFunc.range f), ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x ** congr ** Qed
| |
MeasureTheory.SimpleFunc.integral_eq_sum_of_subset ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s ⊢ integral μ f = ∑ x in s, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x ** rw [SimpleFunc.integral_eq_sum_filter, Finset.sum_subset hs] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s ⊢ ∀ (x : F), x ∈ s → ¬x ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f) → ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x = 0 ** rintro x - hx ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s x : F hx : ¬x ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊢ ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x = 0 ** rw [Finset.mem_filter, not_and_or, Ne.def, Classical.not_not] at hx ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s x : F hx : ¬x ∈ SimpleFunc.range f ∨ x = 0 ⊢ ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x = 0 ** rcases hx.symm with (rfl | hx) ** case inr α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s x : F hx✝ : ¬x ∈ SimpleFunc.range f ∨ x = 0 hx : ¬x ∈ SimpleFunc.range f ⊢ ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x = 0 ** rw [SimpleFunc.mem_range] at hx ** case inr α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s x : F hx✝ : ¬x ∈ SimpleFunc.range f ∨ x = 0 hx : ¬x ∈ Set.range ↑f ⊢ ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x = 0 ** simp only [Set.mem_range, not_exists] at hx ** case inr α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s x : F hx✝ : ¬x ∈ SimpleFunc.range f ∨ x = 0 hx : ∀ (x_1 : α), ¬↑f x_1 = x ⊢ ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • x = 0 ** rw [preimage_eq_empty] <;> simp [Set.disjoint_singleton_left, hx] ** case inl α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝³ : NormedAddCommGroup G inst✝² : NormedAddCommGroup F' inst✝¹ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝ : DecidablePred fun x => x ≠ 0 f : α →ₛ F s : Finset F hs : filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊆ s hx : ¬0 ∈ SimpleFunc.range f ∨ 0 = 0 ⊢ ENNReal.toReal (↑↑μ (↑f ⁻¹' {0})) • 0 = 0 ** simp ** Qed
| |
MeasureTheory.SimpleFunc.integral_piecewise_zero ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s ⊢ integral μ (piecewise s hs f 0) = integral (Measure.restrict μ s) f ** refine' (integral_eq_sum_of_subset _).trans
((sum_congr rfl fun y hy => _).trans (integral_eq_sum_filter _ _).symm) ** case refine'_1 α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s ⊢ filter (fun x => x ≠ 0) (SimpleFunc.range (piecewise s hs f 0)) ⊆ filter (fun x => x ≠ 0) (SimpleFunc.range f) ** intro y hy ** case refine'_1 α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s y : F hy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range (piecewise s hs f 0)) ⊢ y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f) ** simp only [mem_filter, mem_range, coe_piecewise, coe_zero, piecewise_eq_indicator,
mem_range_indicator] at * ** case refine'_1 α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s y : F hy : (y = 0 ∧ s ≠ Set.univ ∨ y ∈ ↑f '' s) ∧ y ≠ 0 ⊢ y ∈ Set.range ↑f ∧ y ≠ 0 ** rcases hy with ⟨⟨rfl, -⟩ | ⟨x, -, rfl⟩, h₀⟩ ** case refine'_1.intro.inl.intro α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s h₀ : 0 ≠ 0 ⊢ 0 ∈ Set.range ↑f ∧ 0 ≠ 0 case refine'_1.intro.inr.intro.intro α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s x : α h₀ : ↑f x ≠ 0 ⊢ ↑f x ∈ Set.range ↑f ∧ ↑f x ≠ 0 ** exacts [(h₀ rfl).elim, ⟨Set.mem_range_self _, h₀⟩] ** case refine'_2 α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s y : F hy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊢ ENNReal.toReal (↑↑μ (↑(piecewise s hs f 0) ⁻¹' {y})) • y = ENNReal.toReal (↑↑(Measure.restrict μ s) (↑f ⁻¹' {y})) • y ** dsimp ** case refine'_2 α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s y : F hy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊢ ENNReal.toReal (↑↑μ (Set.piecewise s (↑f) 0 ⁻¹' {y})) • y = ENNReal.toReal (↑↑(Measure.restrict μ s) (↑f ⁻¹' {y})) • y ** rw [Set.piecewise_eq_indicator, indicator_preimage_of_not_mem,
Measure.restrict_apply (f.measurableSet_preimage _)] ** case refine'_2.ht α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m✝ : MeasurableSpace α μ✝ : Measure α m : MeasurableSpace α f : α →ₛ F μ : Measure α s : Set α hs : MeasurableSet s y : F hy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f) ⊢ ¬0 ∈ {y} ** exact fun h₀ => (mem_filter.1 hy).2 (Eq.symm h₀) ** Qed
| |
MeasureTheory.SimpleFunc.integral_eq_lintegral' ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ ⊢ integral μ (map (ENNReal.toReal ∘ g) f) = ENNReal.toReal (∫⁻ (a : α), g (↑f a) ∂μ) ** have hf' : f.FinMeasSupp μ := integrable_iff_finMeasSupp.1 hf ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ ⊢ integral μ (map (ENNReal.toReal ∘ g) f) = ENNReal.toReal (∫⁻ (a : α), g (↑f a) ∂μ) ** simp only [← map_apply g f, lintegral_eq_lintegral] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ ⊢ integral μ (map (ENNReal.toReal ∘ g) f) = ENNReal.toReal (lintegral (map g f) μ) ** rw [map_integral f _ hf, map_lintegral, ENNReal.toReal_sum] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ ⊢ ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • (ENNReal.toReal ∘ g) x = ∑ a in SimpleFunc.range f, ENNReal.toReal (g a * ↑↑μ (↑f ⁻¹' {a})) ** refine' Finset.sum_congr rfl fun b _ => _ ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ b : E x✝ : b ∈ SimpleFunc.range f ⊢ ENNReal.toReal (↑↑μ (↑f ⁻¹' {b})) • (ENNReal.toReal ∘ g) b = ENNReal.toReal (g b * ↑↑μ (↑f ⁻¹' {b})) ** rw [smul_eq_mul, toReal_mul, mul_comm, Function.comp_apply] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ ⊢ ∀ (a : E), a ∈ SimpleFunc.range f → g a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** rintro a - ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ a : E ⊢ g a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** by_cases a0 : a = 0 ** case pos α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ a : E a0 : a = 0 ⊢ g a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** rw [a0, hg0, zero_mul] ** case pos α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ a : E a0 : a = 0 ⊢ 0 ≠ ⊤ ** exact WithTop.zero_ne_top ** case neg α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ a : E a0 : ¬a = 0 ⊢ g a * ↑↑μ (↑f ⁻¹' {a}) ≠ ⊤ ** apply mul_ne_top (ht a) (hf'.meas_preimage_singleton_ne_zero a0).ne ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝² : NormedAddCommGroup G inst✝¹ : NormedAddCommGroup F' inst✝ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α f : α →ₛ E g : E → ℝ≥0∞ hf : Integrable ↑f hg0 : g 0 = 0 ht : ∀ (b : E), g b ≠ ⊤ hf' : SimpleFunc.FinMeasSupp f μ ⊢ (ENNReal.toReal ∘ g) 0 = 0 ** simp [hg0] ** Qed
| |
MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_integral_norm ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁹ : NormedAddCommGroup E inst✝⁸ : NormedAddCommGroup F inst✝⁷ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedAddCommGroup F' inst✝⁴ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedSpace ℝ E inst✝ : SMulCommClass ℝ 𝕜 E T : Set α → E →L[ℝ] F C : ℝ hT_norm : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s) f : α →ₛ E hf : Integrable ↑f ⊢ C * ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖ = C * integral μ (map norm f) ** rw [map_integral f norm hf norm_zero] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁹ : NormedAddCommGroup E inst✝⁸ : NormedAddCommGroup F inst✝⁷ : NormedSpace ℝ F p : ℝ≥0∞ G : Type u_5 F' : Type u_6 inst✝⁶ : NormedAddCommGroup G inst✝⁵ : NormedAddCommGroup F' inst✝⁴ : NormedSpace ℝ F' m : MeasurableSpace α μ : Measure α inst✝³ : NormedField 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedSpace ℝ E inst✝ : SMulCommClass ℝ 𝕜 E T : Set α → E →L[ℝ] F C : ℝ hT_norm : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ‖T s‖ ≤ C * ENNReal.toReal (↑↑μ s) f : α →ₛ E hf : Integrable ↑f ⊢ C * ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) * ‖x‖ = C * ∑ x in SimpleFunc.range f, ENNReal.toReal (↑↑μ (↑f ⁻¹' {x})) • ‖x‖ ** simp_rw [smul_eq_mul] ** Qed
| |
MeasureTheory.L1.SimpleFunc.norm_eq_integral ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F m : MeasurableSpace α μ : Measure α f : { x // x ∈ simpleFunc E 1 μ } ⊢ ‖f‖ = SimpleFunc.integral μ (SimpleFunc.map norm (toSimpleFunc f)) ** rw [norm_eq_sum_mul f, (toSimpleFunc f).map_integral norm (SimpleFunc.integrable f) norm_zero] ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝¹ : NormedAddCommGroup E inst✝ : NormedAddCommGroup F m : MeasurableSpace α μ : Measure α f : { x // x ∈ simpleFunc E 1 μ } ⊢ ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖ = ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) • ‖x‖ ** simp_rw [smul_eq_mul] ** Qed
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.