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a458723 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import Mathlib
import OAI.Analysis.Conductivity.Model
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
open Matrix Filter Topology
open Set MeasureTheory Filter Topology
open scoped BigOperators
open Set MeasureTheory Filter Topology
open scoped Manifold
open Set Filter
open scoped Topology
open Set Filter MeasureTheory
open scoped Topology Manifold ENNReal
open Set
namespace ScalarConductivity
open Set MeasureTheory Metric
open scoped ENNReal Topology
def openDisjointify {E : Type*} [TopologicalSpace E] {n : ℕ}
(U : Fin n → Set E) (i : Fin n) : Set E :=
U i \ ⋃ j ∈ Finset.Iio i, closure (U j)
lemma openDisjointify_subset {E : Type*} [TopologicalSpace E] {n : ℕ}
(U : Fin n → Set E) (i : Fin n) : openDisjointify U i ⊆ U i := sdiff_subset
lemma openDisjointify_isOpen {E : Type*} [TopologicalSpace E] {n : ℕ}
(U : Fin n → Set E) (hU : ∀ i, IsOpen (U i)) (i : Fin n) :
IsOpen (openDisjointify U i) := by
apply (hU i).sdiff
exact isClosed_biUnion_finset (fun _ _ => isClosed_closure)
lemma openDisjointify_pairwise {E : Type*} [TopologicalSpace E] {n : ℕ}
(U : Fin n → Set E) : Pairwise (fun i j => Disjoint (openDisjointify U i) (openDisjointify U j)) := by
intro i j hij
apply disjoint_left.mpr
intro x hxi hxj
rcases lt_or_gt_of_ne hij with hij | hji
· exact hxj.2 (mem_iUnion₂.mpr ⟨i, Finset.mem_Iio.mpr hij, subset_closure hxi.1⟩)
· exact hxi.2 (mem_iUnion₂.mpr ⟨j, Finset.mem_Iio.mpr hji, subset_closure hxj.1⟩)
lemma openDisjointify_cover {E : Type*} [TopologicalSpace E] {n : ℕ}
(U : Fin n → Set E) (hU : ∀ i, IsOpen (U i)) :
(⋃ i, U i) \ (⋃ i, openDisjointify U i) ⊆ ⋃ i, frontier (U i) := by
classical
intro x hx
by_contra hn
obtain ⟨j, hj⟩ := mem_iUnion.mp hx.1
let s : Finset (Fin n) := Finset.univ.filter (fun i => x ∈ U i)
have hsn : s.Nonempty := ⟨j, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hj⟩⟩
let i := s.min' hsn
have hi : x ∈ U i := (Finset.mem_filter.mp (Finset.min'_mem s hsn)).2
apply hx.2
apply mem_iUnion.mpr
refine ⟨i, hi, ?_⟩
intro hc
obtain ⟨j, hj, hxj⟩ := mem_iUnion₂.mp hc
have hfront : x ∉ frontier (U j) := fun h => hn (mem_iUnion.mpr ⟨j, h⟩)
have hin : x ∈ U j := by
by_contra hnot
apply hfront
rw [frontier, (hU j).interior_eq]
exact ⟨hxj, hnot⟩
have hmin : i ≤ j := Finset.min'_le s j (Finset.mem_filter.mpr ⟨Finset.mem_univ _, hin⟩)
exact not_lt_of_ge hmin (Finset.mem_Iio.mp hj)
lemma openDisjointify_ae_cover {E : Type*} [TopologicalSpace E] [MeasurableSpace E]
(μ : Measure E) {n : ℕ} (U : Fin n → Set E) (hU : ∀ i, IsOpen (U i))
(hf : ∀ i, μ (frontier (U i)) = 0) :
μ ((⋃ i, U i) \ ⋃ i, openDisjointify U i) = 0 := by
exact measure_mono_null (openDisjointify_cover U hU) (measure_iUnion_null hf)
theorem finite_disjoint_localization
{E I : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E] [BorelSpace E]
(μ : Measure E) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ]
{A : Set E} (hA : MeasurableSet A) (hμA : μ A ≠ ∞)
(U : I → Set E) (hU : ∀ i, IsOpen (U i)) (hc : A ⊆ ⋃ i, U i)
{ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ (n : ℕ) (Q : Fin n → Set E) (tag : Fin n → I),
(∀ i, IsOpen (Q i) ∧ IsCompact (closure (Q i)) ∧ closure (Q i) ⊆ U (tag i)) ∧
Pairwise (fun i j => Disjoint (Q i) (Q j)) ∧ μ (A \ ⋃ i, Q i) < ε := by
classical
obtain ⟨K, hKA, hK, hKμ⟩ := hA.exists_isCompact_sdiff_lt hμA hε
have hb : ∀ x : K, ∃ i : I, ∃ r : ℝ, 0 < r ∧ closedBall x.val r ⊆ U i := by
intro x
obtain ⟨i, hi⟩ := mem_iUnion.mp (hc (hKA x.property))
obtain ⟨r, hr, hri⟩ := Metric.isOpen_iff.mp (hU i) x.val hi
refine ⟨i, r/2, by positivity, ?_⟩
intro y hy
exact hri (show dist y x.val < r from (mem_closedBall.mp hy).trans_lt (by linarith))
choose tag r hr hri using hb
obtain ⟨t, ht⟩ := hK.elim_finite_subcover (fun x : K => Metric.ball x.val (r x))
(fun _ => isOpen_ball) (fun x hx => mem_iUnion.mpr ⟨⟨x,hx⟩, mem_ball_self (hr ⟨x,hx⟩)⟩)
let T := {x : K // x ∈ t}
let n := Fintype.card T
let e : Fin n ≃ T := (Fintype.equivFin T).symm
let B : Fin n → Set E := fun i => Metric.ball (e i).val.val (r (e i).val)
have hKB : K ⊆ ⋃ i, B i := by
intro x hx
obtain ⟨y, hyt, hxy⟩ := mem_iUnion₂.mp (ht hx)
let z : T := ⟨y, hyt⟩
exact mem_iUnion.mpr ⟨e.symm z, by simpa only [B, Equiv.apply_symm_apply] using hxy⟩
have hBo : ∀ i, IsOpen (B i) := fun _ => isOpen_ball
have hBf : ∀ i, μ (frontier (B i)) = 0 := by
intro i
exact measure_mono_null frontier_ball_subset_sphere (Measure.addHaar_sphere μ _ _)
refine ⟨n, openDisjointify B, fun i => tag (e i).val, ?_, openDisjointify_pairwise B, ?_⟩
· intro i
have hcl : closure (openDisjointify B i) ⊆ closedBall (e i).val.val (r (e i).val) :=
(closure_mono (openDisjointify_subset B i)).trans closure_ball_subset_closedBall
exact ⟨openDisjointify_isOpen B hBo i,
(isCompact_closedBall _ _).of_isClosed_subset isClosed_closure hcl, hcl.trans (hri (e i).val)⟩
· have hsub : A \ (⋃ i, openDisjointify B i) ⊆
(A \ K) ∪ ((⋃ i, B i) \ ⋃ i, openDisjointify B i) := by
intro x hx
by_cases hxK : x ∈ K
· exact Or.inr ⟨hKB hxK, hx.2⟩
· exact Or.inl ⟨hx.1, hxK⟩
calc
μ (A \ ⋃ i, openDisjointify B i) ≤
μ (A \ K) + μ ((⋃ i, B i) \ ⋃ i, openDisjointify B i) :=
(measure_mono hsub).trans (measure_union_le _ _)
_ = μ (A \ K) := by rw [openDisjointify_ae_cover μ B hBo hBf, add_zero]
_ < ε := hKμ
end ScalarConductivity
end
end OAI
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