| import Mathlib |
| import OAI.Geometry.IntegralFillings.Differentiation.LargePieces |
|
|
| 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] |
| {X : Type*} [MetricSpace X] [SeparableSpace X] [Nonempty X] |
| {f : E → X} {K : ℝ≥0} |
| lemma exists_radius_uniform_metric_error |
| (hf : LipschitzWith K f) {s : Set E} {x : E} |
| (hc : ∀ j : ℕ, ContinuousWithinAt (fun y => metricSeminorm f y (denseSeq E j)) s x) |
| {r : ℕ → ℝ} (hr : ∀ n, 0 < r n) |
| (hu : TendstoUniformlyOn (fun n => metricErrorModulus f (r n)) (fun _ => 0) atTop s) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ δ > 0, ∀ y ∈ s ∩ ball x δ, ∀ z ∈ s ∩ ball x δ, |
| |dist (f y) (f z)-metricSeminorm f x (y-z)| ≤ ε*‖y-z‖ := by |
| have hp := eventually_seminorm_bound_of_dense (fun y => metricSeminorm_lipschitz hf y) |
| (metricSeminorm_lipschitz hf x) (denseRange_denseSeq E) hc (half_pos hε) |
| obtain ⟨a,ha,hap⟩ := Metric.mem_nhdsWithin_iff.1 hp |
| obtain ⟨N,hN⟩ := (eventually_atTop.1 (Metric.tendstoUniformlyOn_iff.1 hu (ε/2) (half_pos hε))) |
| have hNN := hN N le_rfl |
| refine ⟨min a (r N/3), lt_min ha (div_pos (hr N) (by norm_num)), ?_⟩ |
| intro y hy z hz |
| have hyp : ∀ v : E, |metricSeminorm f y v-metricSeminorm f x v| ≤ ε/2*‖v‖ := |
| hap ⟨show dist y x < a from lt_of_lt_of_le hy.2 (min_le_left _ _),hy.1⟩ |
| have hyz : ‖z-y‖ < r N := by |
| have h₁ : ‖y-x‖ < r N/3 := by |
| exact lt_of_lt_of_le (by simpa only [mem_ball,dist_eq_norm] using hy.2) (min_le_right _ _) |
| have h₂ : ‖z-x‖ < r N/3 := by |
| exact lt_of_lt_of_le (by simpa only [mem_ball,dist_eq_norm] using hz.2) (min_le_right _ _) |
| have h₃ : ‖z-y‖ ≤ ‖z-x‖+‖y-x‖ := by |
| calc ‖z-y‖ = ‖(z-x)-(y-x)‖ := by congr 1; abel |
| _ ≤ _ := norm_sub_le _ _ |
| linarith [hr N] |
| have he₁ := metricErrorModulus_bound hf y (z-y) hyz |
| have he₂ := hyp (z-y) |
| have hm : metricErrorModulus f (r N) y < ε/2 := by |
| have hm := hNN y hy.1 |
| simpa only [dist_zero_left,Real.norm_eq_abs, |
| abs_of_nonneg (metricErrorModulus_nonneg hf _ _)] using hm |
| have hpneg : metricSeminorm f x (y-z) = metricSeminorm f x (z-y) := by |
| rw [←neg_sub z y,map_neg_eq_map] |
| rw [hpneg,dist_comm,norm_sub_rev] |
| calc |dist (f z) (f y)-metricSeminorm f x (z-y)| ≤ |
| |dist (f z) (f y)-metricSeminorm f y (z-y)| + |
| |metricSeminorm f y (z-y)-metricSeminorm f x (z-y)| := abs_sub_le _ _ _ |
| _ ≤ metricErrorModulus f (r N) y*‖z-y‖+ε/2*‖z-y‖ := by |
| apply add_le_add _ he₂ |
| simpa only [←add_sub_assoc,add_sub_cancel_left] using he₁ |
| _ ≤ ε*‖z-y‖ := by nlinarith [norm_nonneg (z-y)] |
|
|
| lemma exists_radius_nearIsometric |
| (hf : LipschitzWith K f) {s : Set E} {x : E} |
| (hc : ∀ j : ℕ, ContinuousWithinAt (fun y => metricSeminorm f y (denseSeq E j)) s x) |
| {r : ℕ → ℝ} (hr : ∀ n, 0 < r n) |
| (hu : TendstoUniformlyOn (fun n => metricErrorModulus f (r n)) (fun _ => 0) atTop s) |
| {A : ℝ} (hA : 0 < A) (hbound : ∀ v, ‖v‖ ≤ A*metricSeminorm f x v) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ δ > 0, ∀ y ∈ s ∩ ball x δ, ∀ z ∈ s ∩ ball x δ, |
| (1-ε)*metricSeminorm f x (y-z) ≤ dist (f y) (f z) ∧ |
| dist (f y) (f z) ≤ (1+ε)*metricSeminorm f x (y-z) := by |
| obtain ⟨δ,hδ,H⟩ := exists_radius_uniform_metric_error hf hc hr hu (div_pos hε hA) |
| refine ⟨δ,hδ,?_⟩ |
| intro y hy z hz |
| have he : |dist (f y) (f z)-metricSeminorm f x (y-z)| ≤ |
| ε*metricSeminorm f x (y-z) := by |
| apply (H y hy z hz).trans |
| calc (ε/A)*‖y-z‖ ≤ (ε/A)*(A*metricSeminorm f x (y-z)) := |
| mul_le_mul_of_nonneg_left (hbound _) (div_pos hε hA).le |
| _ = ε*metricSeminorm f x (y-z) := by field_simp |
| obtain ⟨hl,hr⟩ := abs_le.1 he |
| constructor <;> linarith |
|
|
| lemma exists_nearIsometric_partition |
| [MeasurableSpace E] [BorelSpace E] {s : Set E} [Nonempty s] |
| (hs : MeasurableSet s) (hf : LipschitzWith K f) |
| (hc : ∀ x ∈ s, ∀ j : ℕ, ContinuousWithinAt |
| (fun y => metricSeminorm f y (denseSeq E j)) s x) |
| {r : ℕ → ℝ} (hr : ∀ n, 0 < r n) |
| (hu : TendstoUniformlyOn (fun n => metricErrorModulus f (r n)) (fun _ => 0) atTop s) |
| {A : ℝ} (hA : 0 < A) (hbound : ∀ x ∈ s, ∀ v, ‖v‖ ≤ A*metricSeminorm f x v) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ (c : ℕ → s) (t : ℕ → Set E), |
| (∀ i, MeasurableSet (t i)) ∧ Pairwise (fun i j => Disjoint (t i) (t j)) ∧ |
| (⋃ i, t i) = s ∧ ∀ 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 |
| choose δ hδ hlocal using fun x : s => |
| exists_radius_nearIsometric hf (hc x x.property) hr hu hA (hbound x x.property) hε |
| obtain ⟨c,hcvr⟩ := (HereditarilyLindelofSpace.isLindelof s).indexed_countable_subcover |
| (fun x : s => ball (x : E) (δ x)) (fun _ => isOpen_ball) (by |
| intro x hx |
| exact mem_iUnion.2 ⟨⟨x,hx⟩,mem_ball_self (hδ ⟨x,hx⟩)⟩) |
| let t : ℕ → Set E := fun i => s ∩ ball (c i : E) (δ (c i)) |
| refine ⟨c,disjointed t, fun i => MeasurableSet.disjointed |
| (fun j => hs.inter isOpen_ball.measurableSet) i,disjoint_disjointed t,?_,?_⟩ |
| · rw [iUnion_disjointed] |
| apply Subset.antisymm |
| · exact iUnion_subset (fun i => inter_subset_left) |
| · intro x hx |
| obtain ⟨i,hi⟩ := mem_iUnion.1 (hcvr hx) |
| exact mem_iUnion.2 ⟨i,hx,hi⟩ |
| · intro i y hy z hz |
| exact hlocal (c i) y (disjointed_le t i hy) z (disjointed_le t i hz) |
|
|
| end SharpIntegralFillings.MetricDifferentiation |
|
|
| 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 |
| (μ : Measure E) [μ.IsAddHaarMeasure] (hf : LipschitzWith K f) |
| {s : Set E} (hs : MeasurableSet s) (hsf : μ s ≠ ∞) |
| {A : ℝ} (hA : 0 < A) (hbound : ∀ x ∈ s, ∀ 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) ∧ |
| ∀ 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 |
| let r : ℕ → ℝ := fun n => 1/((n : ℝ)+1) |
| have hr : ∀ n, 0 < r n := fun n => by dsimp [r]; positivity |
| have hr0 : Tendsto r atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| have hc : ∀ᵐ x ∂μ.restrict s, |
| Tendsto (fun n => metricErrorModulus f (r n) x) atTop (𝓝 0) := by |
| filter_upwards [(ae_hasCenteredMetricDifferential μ hf).filter_mono |
| (ae_mono (Measure.restrict_le_self (μ := μ) (s := s)))] with x hx |
| exact metricErrorModulus_tendsto_zero hf hx hr0 |
| have H (j : ℕ) := exists_large_continuousOn_uniformly μ hs hsf |
| (show Measurable (fun x i => metricSeminorm f x (denseSeq E i)) from |
| Measurable.of_eval (fun index => measurable_metricSeminorm_apply hf (denseSeq E index))) |
| (fun i => measurable_metricErrorModulus hf (r i)) hc (hr j) |
| choose u hus hum hμu hcu huu using H |
| have HP (j : ℕ) : ∃ (c : ℕ → E) (t : ℕ → Set E), |
| (∀ i, MeasurableSet (t i)) ∧ (⋃ i, t i) = u j ∧ |
| (∀ i, (t i).Nonempty → c i ∈ s) ∧ |
| ∀ 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 |
| by_cases hne : (u j).Nonempty |
| · let _ : Nonempty (u j) := hne.to_subtype |
| obtain ⟨c,t,ht,_,htu,hineq⟩ := exists_nearIsometric_partition (hum j) hf |
| (fun x hx i => (continuousOn_pi.1 (hcu j) i) x hx) hr (huu j) hA |
| (fun x hx => hbound x (hus j hx)) hε |
| exact ⟨fun i => c i,t,ht,htu,fun i _ => hus j (c i).property,hineq⟩ |
| · refine ⟨fun _ => 0,fun _ => ∅,fun _ => MeasurableSet.empty,?_,?_,?_⟩ |
| · simp [not_nonempty_iff_eq_empty.mp hne] |
| · simp |
| · simp |
| choose c t htm htunion hcs htineq using HP |
| let e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm |
| let T : ℕ → Set E := fun i => t (e i).1 (e i).2 |
| have hTunion : (⋃ i, T i) = ⋃ j, u j := by |
| ext x |
| simp only [T,mem_iUnion] |
| constructor |
| · rintro ⟨i,hi⟩ |
| exact ⟨(e i).1,htunion (e i).1 ▸ mem_iUnion.2 ⟨(e i).2,hi⟩⟩ |
| · rintro ⟨j,hj⟩ |
| obtain ⟨i,hi⟩ := mem_iUnion.1 ((htunion j).symm ▸ hj) |
| refine ⟨e.symm (j,i),?_⟩ |
| simpa using hi |
| have hu0 : μ (s \ ⋃ j, u j) = 0 := by |
| apply le_antisymm _ bot_le |
| apply ge_of_tendsto (by simpa using ENNReal.tendsto_ofReal hr0) |
| filter_upwards [] with j |
| exact (measure_mono (sdiff_subset_sdiff_right (subset_iUnion u j))).trans (hμu j) |
| refine ⟨fun i => c (e i).1 (e i).2,disjointed T, |
| fun i => MeasurableSet.disjointed (fun k => htm (e k).1 (e k).2) i, |
| ?_,disjoint_disjointed T,?_,?_,?_⟩ |
| · intro i x hx |
| apply hus (e i).1 |
| rw [←htunion (e i).1] |
| exact mem_iUnion.2 ⟨(e i).2,disjointed_le T i hx⟩ |
| · rw [iUnion_disjointed,hTunion] |
| exact hu0 |
| · intro i hi |
| exact hcs _ _ (hi.mono (disjointed_le T i)) |
| · intro i y hy z hz |
| exact htineq _ _ y (disjointed_le T i hy) z (disjointed_le T i hz) |
|
|
| end SharpIntegralFillings.MetricDifferentiation |
| end |
|
|
| end OAI |
|
|
