| import Mathlib |
| import OAI.Geometry.IntegralFillings.Charts.Differentiability |
|
|
| namespace OAI |
|
|
| section |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open Set MeasureTheory Filter |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace SharpIntegralFillings |
| open Set MeasureTheory Filter Metric |
| open scoped ENNReal NNReal Topology Pointwise |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
| lemma lipschitzOn_image_null {s : Set E} {f : E → E} {K : ℝ≥0} |
| (hf : LipschitzOnWith K f s) (hs : μ s = 0) : μ (f '' s) = 0 := by |
| let H : Measure E := Measure.hausdorffMeasure (Module.finrank ℝ E) |
| have ha : H ≪ μ := Measure.absolutelyContinuous_isAddHaarMeasure H μ |
| have hb : μ ≪ H := Measure.absolutelyContinuous_isAddHaarMeasure μ H |
| apply hb |
| apply le_antisymm _ zero_le |
| have hi := hf.hausdorffMeasure_image_le (d := (Module.finrank ℝ E : ℝ)) (by positivity) |
| change H (f '' s) ≤ (K : ℝ≥0∞) ^ (Module.finrank ℝ E : ℝ) * H s at hi |
| simpa only [ha hs,mul_zero] using hi |
|
|
| lemma lipschitzOn_image_ae_congr_subset {s t : Set E} {f : E → E} {K : ℝ≥0} |
| (hf : LipschitzOnWith K f s) (hts : t ⊆ s) (heq : s =ᵐ[μ] t) : |
| f '' s =ᵐ[μ] f '' t := by |
| apply ae_eq_set.mpr |
| constructor |
| · have hsub : f '' s \ f '' t ⊆ f '' (s \ t) := by |
| rintro y ⟨⟨x,hx,rfl⟩,hy⟩ |
| exact ⟨x,⟨hx,fun h => hy ⟨x,h,rfl⟩⟩,rfl⟩ |
| exact measure_mono_null hsub |
| (lipschitzOn_image_null μ (hf.mono sdiff_subset) (ae_eq_set.mp heq).1) |
| · rw [sdiff_eq_empty.mpr (image_mono hts),measure_empty] |
|
|
| lemma exists_measurable_full_derivative_domain {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K : ℝ≥0} (hf : LipschitzOnWith K f s) : |
| ∃ t, t ⊆ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s ∧ |
| ∀ x ∈ t, HasFDerivWithinAt f (fderivWithin ℝ f s x) t x := by |
| let d : Set E := {x | x ∈ s ∧ DifferentiableWithinAt ℝ f s x} |
| have hd : d =ᵐ[μ] s := by |
| filter_upwards [(ae_restrict_iff' hs).mp (hf.ae_differentiableWithinAt hs (μ := μ))] with x hx |
| change (x ∈ s ∧ DifferentiableWithinAt ℝ f s x) = (x ∈ s) |
| exact propext ⟨And.left,fun h => ⟨h,hx h⟩⟩ |
| obtain ⟨t,htd,htm,hteq⟩ := (hs.nullMeasurableSet.congr hd.symm).exists_measurable_subset_ae_eq |
| have hts : t ⊆ s := fun x hx => (htd hx).1 |
| exact ⟨t,hts,htm,hteq.trans hd,fun x hx => (htd hx).2.hasFDerivWithinAt.mono hts⟩ |
| |
| lemma integral_image_eq_integral_abs_det_within {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K : ℝ≥0} (hf : LipschitzOnWith K f s) (hinj : InjOn f s) |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] (g : E → V) : |
| ∫ y in f '' s, g y ∂μ = |
| ∫ x in s, |(fderivWithin ℝ f s x).det| • g (f x) ∂μ := by |
| obtain ⟨t,hts,htm,hteq,htd⟩ := exists_measurable_full_derivative_domain μ hs hf |
| calc |
| _ = ∫ y in f '' t, g y ∂μ := |
| setIntegral_congr_set (lipschitzOn_image_ae_congr_subset μ hf hts hteq.symm) |
| _ = ∫ x in t, |(fderivWithin ℝ f s x).det| • g (f x) ∂μ := |
| integral_image_eq_integral_abs_det_fderiv_smul μ htm htd (hinj.mono hts) g |
| _ = _ := setIntegral_congr_set hteq |
| |
| lemma integrableOn_image_iff_integrableOn_abs_det_within {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K : ℝ≥0} (hf : LipschitzOnWith K f s) (hinj : InjOn f s) |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] (g : E → V) : |
| IntegrableOn g (f '' s) μ ↔ |
| IntegrableOn (fun x => |(fderivWithin ℝ f s x).det| • g (f x)) s μ := by |
| obtain ⟨t,hts,htm,hteq,htd⟩ := exists_measurable_full_derivative_domain μ hs hf |
| exact (integrableOn_congr_set_ae (lipschitzOn_image_ae_congr_subset μ hf hts hteq.symm)).trans |
| ((integrableOn_image_iff_integrableOn_abs_det_fderiv_smul μ htm htd (hinj.mono hts) g).trans |
| (integrableOn_congr_set_ae hteq)) |
| |
| omit [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma lipschitzOn_invFunOn {s : Set E} {f : E → E} {L : ℝ≥0} |
| (ha : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ (L : ℝ) * dist (f x) (f y)) : |
| LipschitzOnWith L (Function.invFunOn f s) (f '' s) := by |
| rw [lipschitzOnWith_iff_dist_le_mul] |
| intro x hx y hy |
| have h := ha _ (Function.invFunOn_mem hx) _ (Function.invFunOn_mem hy) |
| simpa only [Function.invFunOn_eq hx,Function.invFunOn_eq hy] using h |
| |
| omit [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma antilipschitzOn_injOn {s : Set E} {f : E → E} {L : ℝ≥0} |
| (ha : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ (L : ℝ) * dist (f x) (f y)) : |
| InjOn f s := by |
| intro x hx y hy heq |
| have h := ha x hx y hy |
| rw [heq,dist_self,mul_zero] at h |
| exact dist_le_zero.mp h |
| |
| lemma bilipschitzOn_ae_pullback {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K L : ℝ≥0} (hf : LipschitzOnWith K f s) |
| (ha : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ (L : ℝ) * dist (f x) (f y)) |
| {p : E → Prop} (hp : ∀ᵐ y ∂μ.restrict (f '' s), p y) : |
| ∀ᵐ x ∂μ.restrict s, p (f x) := by |
| have hinj := antilipschitzOn_injOn ha |
| have him := hs.image_of_continuousOn_injOn hf.continuousOn hinj |
| have hnull : μ ({y | ¬ p y} ∩ f '' s) = 0 := by |
| rwa [ae_iff,Measure.restrict_apply' him] at hp |
| rw [ae_iff,Measure.restrict_apply' hs] |
| apply measure_mono_null _ |
| (lipschitzOn_image_null μ ((lipschitzOn_invFunOn ha).mono inter_subset_right) hnull) |
| intro x hx |
| refine ⟨f x,⟨hx.1,⟨x,hx.2,rfl⟩⟩,?_⟩ |
| exact hinj.leftInvOn_invFunOn hx.2 |
| |
| lemma ae_fderivWithin_comp_bilipschitzOn {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K L : ℝ≥0} (hf : LipschitzOnWith K f s) |
| (ha : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ (L : ℝ) * dist (f x) (f y)) |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] |
| {g : E → V} {J : ℝ≥0} (hg : LipschitzOnWith J g (f '' s)) : |
| ∀ᵐ x ∂μ.restrict s, fderivWithin ℝ (g ∘ f) s x = |
| (fderivWithin ℝ g (f '' s) (f x)).comp (fderivWithin ℝ f s x) := by |
| have hinj := antilipschitzOn_injOn ha |
| have him := hs.image_of_continuousOn_injOn hf.continuousOn hinj |
| filter_upwards [ae_uniqueDiffWithinAt μ s,hf.ae_differentiableWithinAt hs (μ := μ), |
| bilipschitzOn_ae_pullback μ hs hf ha (hg.ae_differentiableWithinAt him (μ := μ))] |
| with x hx hfx hgx |
| exact fderivWithin_comp x hgx hfx (mapsTo_image f s) hx |
| |
| lemma ae_det_fderivWithin_ne_zero_bilipschitzOn {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K L : ℝ≥0} (hf : LipschitzOnWith K f s) |
| (ha : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ (L : ℝ) * dist (f x) (f y)) : |
| ∀ᵐ x ∂μ.restrict s, (fderivWithin ℝ f s x).det ≠ 0 := by |
| have hinj := antilipschitzOn_injOn ha |
| filter_upwards [ae_fderivWithin_comp_bilipschitzOn μ hs hf ha (lipschitzOn_invFunOn ha), |
| ae_uniqueDiffWithinAt μ s,ae_restrict_mem hs] with x hx hu hxs |
| have heq : EqOn (Function.invFunOn f s ∘ f) id s := |
| fun x hxs => hinj.leftInvOn_invFunOn hxs |
| rw [fderivWithin_congr' heq hxs,fderivWithin_id hu] at hx |
| have hd := congrArg (fun a : E →L[ℝ] E => a.det) hx |
| simp only [ContinuousLinearMap.det, ContinuousLinearMap.toLinearMap_comp, |
| LinearMap.det_comp, ContinuousLinearMap.coe_id, LinearMap.det_id] at hd |
| intro hz |
| change LinearMap.det _ = 0 at hz |
| rw [hz,mul_zero] at hd |
| exact one_ne_zero hd |
|
|
| lemma aemeasurable_fderivWithin_lipschitzOn {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K : ℝ≥0} (hf : LipschitzOnWith K f s) : |
| AEMeasurable (fderivWithin ℝ f s) (μ.restrict s) := by |
| obtain ⟨t,hts,htm,hteq,htd⟩ := exists_measurable_full_derivative_domain μ hs hf |
| have h := aemeasurable_fderivWithin μ htm htd |
| rwa [Measure.restrict_congr_set hteq] at h |
|
|
| lemma ae_norm_fderivWithin_lipschitzOn {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K : ℝ≥0} (hf : LipschitzOnWith K f s) : |
| ∀ᵐ x ∂μ.restrict s, ‖fderivWithin ℝ f s x‖ ≤ K := by |
| obtain ⟨t,hts,htm,hteq,htd⟩ := exists_measurable_full_derivative_domain μ hs hf |
| have hlin : ApproximatesLinearOn f (0 : E →L[ℝ] E) t K := by |
| intro x hx y hy |
| simpa only [_root_.zero_apply,sub_zero,←dist_eq_norm] using |
| hf.dist_le_mul x (hts hx) y (hts hy) |
| have h := hlin.norm_fderiv_sub_le μ htm (fderivWithin ℝ f s) htd |
| rw [Measure.restrict_congr_set hteq] at h |
| filter_upwards [h] with x hx |
| exact_mod_cast (by simpa only [sub_zero] using hx : ‖fderivWithin ℝ f s x‖₊ ≤ K) |
|
|
| lemma ae_bound_det_fderivWithin_lipschitzOn {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K : ℝ≥0} (hf : LipschitzOnWith K f s) : |
| ∃ C : ℝ, ∀ᵐ x ∂μ.restrict s, ‖(fderivWithin ℝ f s x).det‖ ≤ C := by |
| obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : E →L[ℝ] E) (K : ℝ)).exists_bound_of_continuousOn |
| (ContinuousLinearMap.continuous_det (𝕜 := ℝ) (E := E)).continuousOn |
| refine ⟨C,?_⟩ |
| filter_upwards [ae_norm_fderivWithin_lipschitzOn μ hs hf] with x hx |
| exact hC _ (by simpa only [mem_closedBall,dist_zero_right] using hx) |
|
|
| lemma integrable_mul_det_fderivWithin_lipschitzOn {s : Set E} (hs : MeasurableSet s) |
| {f : E → E} {K : ℝ≥0} (hf : LipschitzOnWith K f s) |
| {g : E → ℝ} (hg : IntegrableOn g s μ) : |
| IntegrableOn (fun x => g x * (fderivWithin ℝ f s x).det) s μ := by |
| obtain ⟨C,hC⟩ := ae_bound_det_fderivWithin_lipschitzOn μ hs hf |
| exact hg.mul_bdd ((ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable |
| (aemeasurable_fderivWithin_lipschitzOn μ hs hf)).aestronglyMeasurable) hC |
|
|
| end SharpIntegralFillings |
| end |
|
|
| end OAI |
|
|
