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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | 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
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