| 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 |
| |
