| import Mathlib |
| import OAI.Geometry.IntegralFillings.Charts.Control |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace SharpIntegralFillings |
| open MeasureTheory Set |
|
|
| variable {Z X : Type*} [MeasurableSpace Z] [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
| lemma signed_chart_control_dominates_on |
| (μ : Measure Z) (ν : Measure X) [IsFiniteMeasure ν] |
| {φ : Z → X} (hφ : Measurable φ) {s₀ : Set Z} |
| (hμ : ∀ᵐ z ∂μ, z ∈ s₀) |
| (he : MeasurableEmbedding (fun z : s₀ => φ (z : Z))) {w : Z → ℝ} |
| (hw : Integrable w μ) |
| (hcontrol : ∀ b : X → ℝ, BoundedLip b → |
| |∫ z, w z * b (φ z) ∂μ| ≤ ∫ x, |b x| ∂ν) : |
| densityPush μ φ (fun z => |w z|) ≤ ν := by |
| let w' := hw.aestronglyMeasurable.mk w |
| have heq : w =ᵐ[μ] w' := hw.aestronglyMeasurable.ae_eq_mk |
| have hτ : densityPush μ φ (fun z => |w z|) = densityPush μ φ (fun z => |w' z|) := by |
| simp only [densityPush_eq_map μ hφ] |
| congr 1 |
| apply withDensity_congr_ae |
| exact heq.mono fun z hz => by dsimp only; rw [hz] |
| rw [hτ] |
| apply signed_chart_control_dominates_on_of_measurable μ ν hφ hμ he |
| hw.aestronglyMeasurable.measurable_mk (hw.congr heq) |
| intro b hb |
| have hi : (∫ z, w' z*b (φ z) ∂μ) = ∫ z, w z*b (φ z) ∂μ := |
| integral_congr_ae (heq.mono fun z hz => by dsimp only; rw [hz]) |
| rw [hi] |
| exact hcontrol b hb |
| end SharpIntegralFillings |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
|
|
| namespace SharpIntegralFillings |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [Measure.IsAddHaarMeasure μ] |
|
|
| lemma tangentConeAt_eq_univ_of_density_one (s : Set E) (x : E) |
| (hx : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) |
| (𝓝[>] 0) (𝓝 1)) : tangentConeAt ℝ s x = univ := by |
| apply eq_univ_of_forall |
| intro z |
| rw [tangentConeAt_eq_biInter_closure] |
| simp only [mem_iInter] |
| intro U hU |
| rw [Metric.mem_closure_iff] |
| intro ε hε |
| have B₁ : ∀ᶠ r in 𝓝[>] (0 : ℝ), (s ∩ ({x} + r • closedBall z (ε / 2))).Nonempty := |
| Measure.eventually_nonempty_inter_smul_of_density_one μ s x hx _ measurableSet_closedBall |
| (measure_closedBall_pos μ z (half_pos hε)).ne' |
| have B₂ : ∀ᶠ r in 𝓝[>] (0 : ℝ), r • closedBall z (ε / 2) ⊆ U := by |
| apply nhdsWithin_le_nhds |
| have h := eventually_singleton_add_smul_subset (𝕜 := ℝ) (x := (0 : E)) |
| (s := closedBall z (ε / 2)) isBounded_closedBall hU |
| filter_upwards [h] with r hr |
| simpa using hr |
| obtain ⟨r, ⟨y, ys, hy⟩, hrU, hr⟩ := |
| (B₁.and (B₂.and self_mem_nhdsWithin)).exists |
| obtain ⟨a, ha, hya⟩ : ∃ a, a ∈ closedBall z (ε / 2) ∧ y = x + r • a := by |
| simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy |
| rcases hy with ⟨a, ha, h⟩ |
| exact ⟨a, ha, by simp only [h, add_neg_cancel_left]⟩ |
| refine ⟨a, ?_, ?_⟩ |
| · have hr0 : r ≠ 0 := ne_of_gt hr |
| have hra : r • a ∈ U ∩ (x + ·) ⁻¹' s := |
| ⟨hrU (smul_mem_smul_set ha), by simpa only [mem_preimage, ← hya] using ys⟩ |
| exact ⟨r⁻¹, mem_univ _, r • a, hra, inv_smul_smul₀ hr0 a⟩ |
| · exact lt_of_le_of_lt (by simpa [dist_comm] using ha) (half_lt_self hε) |
|
|
| lemma uniqueDiffWithinAt_of_density_one (s : Set E) (x : E) |
| (hx : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) |
| (𝓝[>] 0) (𝓝 1)) : UniqueDiffWithinAt ℝ s x := by |
| have ht := tangentConeAt_eq_univ_of_density_one μ s x hx |
| rw [uniqueDiffWithinAt_iff, ht] |
| exact ⟨by simp, mem_closure_of_nonempty_tangentConeAt (by rw [ht]; exact univ_nonempty)⟩ |
|
|
| lemma ae_uniqueDiffWithinAt (s : Set E) : |
| ∀ᵐ x ∂μ.restrict s, UniqueDiffWithinAt ℝ s x := by |
| filter_upwards [Besicovitch.ae_tendsto_measure_inter_div μ s] with x hx |
| exact uniqueDiffWithinAt_of_density_one μ s x hx |
|
|
| lemma ae_fderivWithin_eq_fderiv_extension {s : Set E} (hs : MeasurableSet s) |
| {f F : E → ℝ} {L : ℝ≥0} (hF : LipschitzWith L F) (heq : EqOn f F s) : |
| ∀ᵐ x ∂μ.restrict s, fderivWithin ℝ f s x = fderiv ℝ F x := by |
| filter_upwards [ae_uniqueDiffWithinAt μ s, ae_restrict_mem hs, |
| (hF.ae_differentiableAt (μ := μ)).filter_mono (ae_mono Measure.restrict_le_self)] |
| with x hx hxs hFx |
| rw [fderivWithin_congr' heq hxs, fderivWithin_eq_fderiv hx hFx] |
|
|
| lemma ae_norm_fderivWithin_le {s : Set E} (hs : MeasurableSet s) |
| {f : E → ℝ} {L : ℝ≥0} (hf : LipschitzOnWith L f s) : |
| ∀ᵐ x ∂μ.restrict s, ‖fderivWithin ℝ f s x‖ ≤ L := by |
| obtain ⟨F, hF, heq⟩ := hf.extend_real |
| filter_upwards [ae_fderivWithin_eq_fderiv_extension μ hs hF heq] with x hx |
| rw [hx] |
| exact norm_fderiv_le_of_lipschitz ℝ hF |
|
|
| end SharpIntegralFillings |
|
|
| end |
|
|
| end OAI |
|
|
