| import OAI.Geometry.RCD.Transport.MeetingPairCollapse |
| import OAI.Geometry.RCD.Heat.ConditionalPairInverse |
|
|
| namespace OAI |
|
|
| /-! # Measurable inversion of interior evaluation for the actual compressed plan -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped ENNReal NNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates.NoncollapsedRCD |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] [SeparableSpace X] {K : ℝ} {n : ℕ} |
| [IsLocallyFiniteMeasure (hausdorff n X)] (hX : NoncollapsedRCD K n X) |
| include hX |
| attribute [local instance] pathBorel pathBorelSpace |
|
|
| theorem compressed_evaluation_inverse (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R) |
| (π : Measure C(Time,X)) [IsProbabilityMeasure π] (hg : ∀ᵐ γ ∂π, IsGeodesic γ) |
| (hp : IsOptimalCoupling (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1)) |
| (π.map (fun γ => (γ 0,γ 1)))) |
| {C : ℝ≥0} (hC : 0 < C) |
| (hc : ∀ t : Time, π.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X) |
| (hs₀ : ∀ᵐ x ∂π.map (fun γ => γ 0), dist x p ≤ R) |
| (hs₁ : ∀ᵐ x ∂π.map (fun γ => γ 1), dist x p ≤ R) |
| (t : Time) (ht0 : 0 < (t:ℝ)) (ht1 : (t:ℝ) < 1) : |
| ∃ g : X → C(Time,X), Measurable g ∧ ∀ᵐ γ ∂π, g (γ t)=γ := by |
| let : CompleteSpace X := hX.complete |
| let : Nonempty C(Time,X) := nonempty_of_isProbabilityMeasure π |
| have hac := Measure.absolutelyContinuous_of_le_smul (hc 0) |
| obtain ⟨u,φ,ψ,L,hφ,hψ,_,_,_,had,hcontact,_⟩ := hp.bounded_sobolev_potential |
| (hausdorff n X) hX.quadratic hX.geodesic _ _ hac p hR hs₀ hs₁ |
| have hct : ∀ᵐ γ ∂π, φ (γ 0)+ψ (γ 1)=dist (γ 0) (γ 1)^2/2 := |
| ae_of_ae_map ((measurable_path_eval 0).prodMk (measurable_path_eval 1)).aemeasurable hcontact |
| let σ := conditionalSelfPair π (fun γ => γ t) (measurable_path_eval t) |
| obtain ⟨hσ₀,hσ₁⟩ := conditionalSelfPair_marginals π (fun γ => γ t) (measurable_path_eval t) |
| let : IsProbabilityMeasure σ := ⟨by |
| have h := congrArg (fun m : Measure C(Time,X) => m univ) hσ₀ |
| simpa only [Measure.map_apply measurable_fst MeasurableSet.univ,preimage_univ,measure_univ] using h⟩ |
| have hmeet : ∀ᵐ z ∂σ, z.1 t=z.2 t := |
| conditionalSelfPair_common π (fun γ => γ t) (measurable_path_eval t) |
| have hdiag := hX.contacting_pair_collapse hn p hR π hg hp hC hc hs₀ hs₁ |
| hφ.continuous.measurable hψ.continuous.measurable had hct σ hσ₀ hσ₁ t ht0 ht1 hmeet |
| exact measurable_inverse_of_conditional_diagonal π (fun γ => γ t) (measurable_path_eval t) hdiag |
|
|
| theorem compressed_evaluation_injective (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R) |
| (π : Measure C(Time,X)) [IsProbabilityMeasure π] (hg : ∀ᵐ γ ∂π, IsGeodesic γ) |
| (hp : IsOptimalCoupling (π.map (fun γ => γ 0)) (π.map (fun γ => γ 1)) |
| (π.map (fun γ => (γ 0,γ 1)))) |
| {C : ℝ≥0} (hC : 0 < C) |
| (hc : ∀ t : Time, π.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X) |
| (hs₀ : ∀ᵐ x ∂π.map (fun γ => γ 0), dist x p ≤ R) |
| (hs₁ : ∀ᵐ x ∂π.map (fun γ => γ 1), dist x p ≤ R) |
| (t : Time) (ht0 : 0 < (t:ℝ)) (ht1 : (t:ℝ) < 1) : |
| ∃ S : Set C(Time,X), MeasurableSet S ∧ (∀ᵐ γ ∂π, γ ∈ S) ∧ InjOn (fun γ => γ t) S := by |
| obtain ⟨g,hg,hgπ⟩ := hX.compressed_evaluation_inverse hn p hR π hg hp hC hc hs₀ hs₁ t ht0 ht1 |
| refine ⟨{γ | g (γ t)=γ},measurableSet_eq_fun (hg.comp (measurable_path_eval t)) measurable_id,hgπ,?_⟩ |
| intro γ hγ ζ hζ he |
| exact hγ.symm.trans ((congrArg g he).trans hζ) |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
|
|
| end |
|
|
| end OAI |
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