| import Mathlib |
| import OAI.Geometry.IntegralFillings.Differentiation.HilbertCharts |
| import OAI.Geometry.IntegralFillings.Differentiation.Partitions |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| 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 |
|
|
| namespace SharpIntegralFillings.MetricDifferentiation |
| open Metric Asymptotics |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [MeasurableSpace E] [BorelSpace E] |
| {X : Type*} [MetricSpace X] [SeparableSpace X] [Nonempty X] |
| {f : E → X} {K : ℝ≥0} |
| theorem exists_ae_nearIsometric_partition_with_property |
| (μ : Measure E) [μ.IsAddHaarMeasure] (hf : LipschitzWith K f) |
| {s : Set E} (hs : MeasurableSet s) (hsf : μ s ≠ ∞) |
| {P : E → Prop} (hP : ∀ᵐ x ∂μ.restrict s, P x) |
| {A : ℝ} (hA : 0 < A) |
| (hbound : ∀ x ∈ s, P x → ∀ v, ‖v‖ ≤ A*metricSeminorm f x v) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ (c : ℕ → E) (t : ℕ → Set E), |
| (∀ i, MeasurableSet (t i)) ∧ (∀ i, t i ⊆ s) ∧ |
| Pairwise (fun i j => Disjoint (t i) (t j)) ∧ μ (s \ ⋃ i, t i) = 0 ∧ |
| (∀ i, (t i).Nonempty → c i ∈ s ∧ P (c i)) ∧ |
| ∀ i, ∀ y ∈ t i, ∀ z ∈ t i, |
| (1-ε)*metricSeminorm f (c i) (y-z) ≤ dist (f y) (f z) ∧ |
| dist (f y) (f z) ≤ (1+ε)*metricSeminorm f (c i) (y-z) := by |
| classical |
| have hnull : μ {x | ¬(x ∈ s → P x)} = 0 := |
| ae_iff.mp ((ae_restrict_iff' hs).mp hP) |
| obtain ⟨N,hN,hNm,hN0⟩ := exists_measurable_superset_of_null hnull |
| let s₀ := s \ N |
| have hs₀ : MeasurableSet s₀ := hs.diff hNm |
| have hprop : ∀ x ∈ s₀, P x := by |
| intro x hx |
| by_contra hp |
| exact hx.2 (hN (by simpa using And.intro hx.1 hp)) |
| obtain ⟨c,t,htm,hts,hdisj,ht0,hcs,hineq⟩ := |
| exists_ae_nearIsometric_partition μ hf hs₀ |
| ((measure_mono sdiff_subset).trans_lt hsf.lt_top).ne hA |
| (fun x hx => hbound x hx.1 (hprop x hx)) hε |
| refine ⟨c,t,htm,fun i => (hts i).trans sdiff_subset,hdisj,?_,?_,hineq⟩ |
| · apply le_antisymm _ bot_le |
| calc μ (s \ ⋃ i, t i) ≤ μ (N ∪ (s₀ \ ⋃ i, t i)) := by |
| apply measure_mono |
| intro x hx |
| by_cases hn : x ∈ N |
| · exact Or.inl hn |
| · exact Or.inr ⟨⟨hx.1,hn⟩,hx.2⟩ |
| _ ≤ μ N + μ (s₀ \ ⋃ i, t i) := measure_union_le _ _ |
| _ = 0 := by rw [hN0,ht0,add_zero] |
| · intro i hi |
| exact ⟨(hcs i hi).1,hprop _ (hcs i hi)⟩ |
| |
| end SharpIntegralFillings.MetricDifferentiation |
| |
| namespace SharpIntegralFillings.MetricDifferentiation |
| open Metric |
| |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| {X : Type*} [MetricSpace X] |
| |
| theorem exists_hilbertian_nearIsometric_chart_partition |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
| {s : Set E} (hs : MeasurableSet s) (hsf : μ s ≠ ∞) {f : s → X} {K J : ℝ≥0} |
| (hf : LipschitzWith K f) (hJ : AntilipschitzWith J f) |
| (hQ : ∀ a b c d : s, |
| dist (f a) (f c)^2+dist (f b) (f d)^2 ≤ |
| dist (f a) (f b)^2+dist (f b) (f c)^2+dist (f c) (f d)^2+dist (f d) (f a)^2) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ (p : ℕ → Seminorm ℝ E) (t : ℕ → Set E), |
| (∀ i, MeasurableSet (t i)) ∧ (∀ i, t i ⊆ s) ∧ |
| Pairwise (fun i j => Disjoint (t i) (t j)) ∧ μ (s \ ⋃ i, t i) = 0 ∧ |
| (∀ i, LipschitzWith K (p i)) ∧ |
| (∀ i, (t i).Nonempty → |
| (∀ u v, p i (u+v)^2+p i (u-v)^2 = 2*p i u^2+2*p i v^2) ∧ |
| (∀ v, ‖v‖ ≤ (J : ℝ)*p i v)) ∧ |
| ∀ i, ∀ y : s, (y : E) ∈ t i → ∀ z : s, (z : E) ∈ t i → |
| (1-ε)*p i ((y : E)-(z : E)) ≤ dist (f y) (f z) ∧ |
| dist (f y) (f z) ≤ (1+ε)*p i ((y : E)-(z : E)) := by |
| classical |
| let Y := Set.range f |
| let : SeparableSpace Y := (isSeparable_range hf.continuous).separableSpace |
| let F : s → Y := fun x => ⟨f x,x,rfl⟩ |
| let e := kuratowskiEmbedding Y |
| have he : Isometry e := kuratowskiEmbedding.isometry Y |
| let F₀ : E → ↥(lp (fun _ : ℕ => ℝ) (⊤ : ENNReal)) := |
| fun x => if hx : x ∈ s then e (F ⟨x,hx⟩) else 0 |
| have hF₀ : LipschitzOnWith K F₀ s := by |
| apply LipschitzOnWith.of_dist_le_mul |
| intro x hx y hy |
| dsimp only [F₀] |
| rw [dite_eq_left hx,dite_eq_left hy,he.dist_eq] |
| exact hf.dist_le_mul ⟨x,hx⟩ ⟨y,hy⟩ |
| obtain ⟨g,hg,heq⟩ := hF₀.extend_lp_infty |
| let Z := Set.range g |
| let : SeparableSpace Z := (isSeparable_range hg.continuous).separableSpace |
| let : Nonempty Z := ⟨⟨g 0,0,rfl⟩⟩ |
| let G : E → Z := fun x => ⟨g x,x,rfl⟩ |
| have hG : LipschitzWith K G := by |
| apply LipschitzWith.of_dist_le_mul |
| exact hg.dist_le_mul |
| have hdist (y z : s) : dist (G y) (G z) = dist (f y) (f z) := by |
| change dist (g y) (g z) = _ |
| rw [←heq y.property, ←heq z.property] |
| dsimp only [F₀] |
| rw [dite_eq_left y.property,dite_eq_left z.property] |
| exact he.dist_eq (F y) (F z) |
| have hQG : ∀ a ∈ s, ∀ b ∈ s, ∀ c ∈ s, ∀ d ∈ s, |
| dist (G a) (G c)^2+dist (G b) (G d)^2 ≤ |
| dist (G a) (G b)^2+dist (G b) (G c)^2+dist (G c) (G d)^2+dist (G d) (G a)^2 := by |
| intro a ha b hb c hc d hd |
| simpa only [← hdist] using hQ ⟨a,ha⟩ ⟨b,hb⟩ ⟨c,hc⟩ ⟨d,hd⟩ |
| have hJG : ∀ a ∈ s, ∀ b ∈ s, (1 : ℝ)*dist a b ≤ (J : ℝ)*dist (G a) (G b) := by |
| intro a ha b hb |
| have hh := hJ.le_mul_dist ⟨a,ha⟩ ⟨b,hb⟩ |
| change dist a b ≤ (J : ℝ)*dist (f ⟨a,ha⟩) (f ⟨b,hb⟩) at hh |
| simpa only [←hdist,one_mul] using hh |
| let P : E → Prop := fun x => |
| (∀ u v, metricSeminorm G x (u+v)^2+metricSeminorm G x (u-v)^2 = |
| 2*metricSeminorm G x u^2+2*metricSeminorm G x v^2) ∧ |
| (∀ v, ‖v‖ ≤ (J : ℝ)*metricSeminorm G x v) |
| have hP : ∀ᵐ x ∂μ.restrict s, P x := by |
| filter_upwards [ae_metricSeminorm_parallelogram_on μ hG hs hsf hQG, |
| ae_metricSeminorm_distance_bound_on μ hG hs hsf hJG] with x hx hj |
| exact ⟨hx,by simpa only [one_mul] using hj⟩ |
| obtain ⟨c,t,htm,hts,hdisj,ht0,hcs,hineq⟩ := |
| exists_ae_nearIsometric_partition_with_property μ hG hs hsf hP |
| (A := (J : ℝ)+1) (by positivity) |
| (fun x _ hx v => (hx.2 v).trans <| mul_le_mul_of_nonneg_right |
| (by linarith) (apply_nonneg (metricSeminorm G x) v)) hε |
| refine ⟨fun i => metricSeminorm G (c i),t,htm,hts,hdisj,ht0, |
| fun i => metricSeminorm_lipschitz hG (c i),fun i hi => (hcs i hi).2,?_⟩ |
| intro i y hy z hz |
| simpa only [hdist] using hineq i y hy z hz |
| end SharpIntegralFillings.MetricDifferentiation |
| |
| end |
| |
| end OAI |
| |
