| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| namespace WeakMTWTransport |
| variable {B : Type*} [TopologicalSpace B] |
| {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {E : B → Type*} [TopologicalSpace (TotalSpace F E)] |
| [∀ x, NormedAddCommGroup (E x)] [∀ x, InnerProductSpace ℝ (E x)] |
| [FiberBundle F E] [VectorBundle ℝ F E] [IsContinuousRiemannianBundle F E] |
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| lemma continuous_bundle_norm : Continuous (fun z : TotalSpace F E => ‖z.2‖) := by |
| have h : Continuous (fun z : TotalSpace F E => inner ℝ z.2 z.2) := |
| (continuous_id : Continuous (id : TotalSpace F E → _)).inner_bundle continuous_id |
| simpa only [← norm_eq_sqrt_real_inner] using h.sqrt |
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| lemma isCompact_bundle_disk [CompactSpace B] [T2Space B] |
| [FiniteDimensional ℝ F] (R : ℝ) : |
| IsCompact {z : TotalSpace F E | ‖z.2‖ ≤ R} := by |
| classical |
| have hsclosed : IsClosed {z : TotalSpace F E | ‖z.2‖ ≤ R} := |
| isClosed_le continuous_bundle_norm continuous_const |
| by_cases hR : R < 0 |
| · have : {z : TotalSpace F E | ‖z.2‖ ≤ R} = ∅ := by |
| ext z |
| simp only [mem_ofPred_eq,mem_empty_iff_false,iff_false] |
| exact not_le.mpr ((hR.trans_le (norm_nonneg _))) |
| rw [this] |
| exact isCompact_empty |
| have hlocal (x : B) : ∃ C : ℝ, 0 < C ∧ ∃ K : Set B, |
| K ∈ 𝓝 x ∧ IsCompact K ∧ K ⊆ (trivializationAt F E x).baseSet ∧ |
| ∀ y ∈ K, ‖(trivializationAt F E x).continuousLinearMapAt ℝ y‖ < C := by |
| obtain ⟨C,hC,hbound⟩ := eventually_norm_trivializationAt_lt F E x |
| obtain ⟨K,hK,hKs,hKc⟩ := local_compact_nhds |
| (inter_mem ((trivializationAt F E x).open_baseSet.mem_nhds |
| (mem_baseSet_trivializationAt F E x)) hbound) |
| exact ⟨C,hC,K,hK,hKc,fun y hy => (hKs hy).1,fun y hy => (hKs hy).2⟩ |
| choose C hC K hKn hKc hKb hKC using hlocal |
| let L (x : B) : Set (TotalSpace F E) := |
| (trivializationAt F E x).toOpenPartialHomeomorph.symm '' |
| (K x ×ˢ Metric.closedBall (0 : F) (C x * R)) |
| have hLc (x : B) : IsCompact (L x) := by |
| apply ((hKc x).prod (isCompact_closedBall (0 : F) (C x * R))).image_of_continuousOn |
| apply (trivializationAt F E x).toOpenPartialHomeomorph.continuousOn_symm.mono |
| intro y hy |
| change y ∈ (trivializationAt F E x).target |
| exact (trivializationAt F E x).mem_target.mpr (hKb x hy.1) |
| obtain ⟨t,ht⟩ := finite_cover_nhds hKn |
| apply (t.isCompact_biUnion (fun x _ => hLc x)).of_isClosed_subset hsclosed |
| intro z hz |
| have hzm : z.1 ∈ ⋃ x ∈ t, K x := by rw [ht]; trivial |
| obtain ⟨x,hx,hzK⟩ := mem_iUnion₂.mp hzm |
| apply mem_iUnion₂.mpr ⟨x,hx,?_⟩ |
| have hzb := hKb x hzK |
| have hzs : z ∈ (trivializationAt F E x).source := |
| (trivializationAt F E x).mem_source.mpr hzb |
| refine ⟨(trivializationAt F E x) z,?_,?_⟩ |
| · refine ⟨?_,?_⟩ |
| · rwa [Trivialization.coe_fst _ hzs] |
| · rw [Metric.mem_closedBall,dist_zero_right] |
| have he : (trivializationAt F E x z).2 = |
| (trivializationAt F E x).continuousLinearMapAt ℝ z.1 z.2 := by |
| rw [Trivialization.continuousLinearMapAt_apply_of_mem ℝ _ hzb] |
| rw [he] |
| calc |
| _ ≤ ‖(trivializationAt F E x).continuousLinearMapAt ℝ z.1‖ * ‖z.2‖ := |
| ContinuousLinearMap.le_opNorm _ _ |
| _ ≤ C x * R := mul_le_mul (hKC x z.1 hzK).le hz (norm_nonneg _) (hC x).le |
| · exact (trivializationAt F E x).left_inv hzs |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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