| import OAI.Geometry.RCD.Transport.CompressedInitialInverse |
|
|
| namespace OAI |
|
|
| /-! # Measurable inverses for every evaluation, including both endpoints -/ |
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|
| 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_final_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) |
| : |
| ∃ g : X → C(Time,X), Measurable g ∧ ∀ᵐ γ ∂π, g (γ 1)=γ := by |
| let : CompleteSpace X := hX.complete |
| let : Nonempty X := nonempty_of_isProbabilityMeasure (π.map (fun γ => γ 0)) |
| have hfin := coupling_transportCost_ne_top hp.1 hp.2.1 p |
| (integrable_dist_sq_of_ae_bounded _ p hs₀) (integrable_dist_sq_of_ae_bounded _ p hs₁) |
| obtain ⟨_,hgrev,hprev⟩ := geodesic_subpath_law π hg hp hfin 1 0 |
| let πr := π.map (subpath 1 0) |
| have hcomp (r : Time) : πr.map (fun γ => γ r) ≤ ENNReal.ofReal C • hausdorff n X := by |
| rw [show πr=π.map (subpath 1 0) from rfl,subpath_map_eval] |
| exact hc _ |
| have hprev' : IsOptimalCoupling (πr.map (fun γ => γ 0)) (πr.map (fun γ => γ 1)) |
| (πr.map (fun γ => (γ 0,γ 1))) := by |
| simpa only [πr,subpath_map_eval,timeSegment_zero,timeSegment_one] using hprev |
| have hsrev₀ : ∀ᵐ x ∂πr.map (fun γ => γ 0), dist x p ≤ R := by |
| simpa only [πr,subpath_map_eval,timeSegment_zero] using hs₁ |
| have hsrev₁ : ∀ᵐ x ∂πr.map (fun γ => γ 1), dist x p ≤ R := by |
| simpa only [πr,subpath_map_eval,timeSegment_one] using hs₀ |
| obtain ⟨g,hg,hgi⟩ := hX.compressed_initial_inverse hn p hR πr hgrev hprev' hC hcomp hsrev₀ hsrev₁ |
| refine ⟨(subpath 1 0) ∘ g,(subpath_measurable 1 0).comp hg,?_⟩ |
| have hi := ae_of_ae_map (μ:=π) (subpath_measurable 1 0).aemeasurable hgi |
| filter_upwards [hi] with γ hγ |
| change g (γ (timeSegment 1 0 0))=subpath 1 0 γ at hγ |
| rw [timeSegment_zero] at hγ |
| have hγ' : g (γ 1)=subpath 1 0 γ := hγ |
| change subpath 1 0 (g (γ 1))=γ |
| rw [hγ',subpath_reverse_reverse] |
|
|
| theorem compressed_time_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) : |
| ∃ g : X → C(Time,X), Measurable g ∧ ∀ᵐ γ ∂π, g (γ t)=γ := by |
| by_cases h0 : t=0 |
| · subst t |
| exact hX.compressed_initial_inverse hn p hR π hg hp hC hc hs₀ hs₁ |
| by_cases h1 : t=1 |
| · subst t |
| exact hX.compressed_final_inverse hn p hR π hg hp hC hc hs₀ hs₁ |
| exact hX.compressed_evaluation_inverse hn p hR π hg hp hC hc hs₀ hs₁ t |
| (lt_of_le_of_ne t.property.1 (by intro he; exact h0 (Subtype.ext he.symm))) |
| (lt_of_le_of_ne t.property.2 (by intro he; exact h1 (Subtype.ext he))) |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
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| end |
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| end OAI |
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