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e4d9d2b | 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 | import OAI.Geometry.RCD.Transport.CompressedInitialInverse
import OAI.Geometry.RCD.Metric.InverseMeasureUnique
namespace OAI
/-! # Uniqueness among bounded compressed optimal dynamical plans -/
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_plan_unique (hn : 2 ≤ n) (p : X) {R : ℝ} (hR : 0 ≤ R)
{ν₀ ν₁ : Measure X}
(π ρ : Measure C(Time,X)) [IsProbabilityMeasure π] [IsProbabilityMeasure ρ]
(hgπ : ∀ᵐ γ ∂π, IsGeodesic γ) (hgρ : ∀ᵐ γ ∂ρ, IsGeodesic γ)
(hpπ : IsOptimalCoupling ν₀ ν₁ (π.map (fun γ => (γ 0,γ 1))))
(hpρ : IsOptimalCoupling ν₀ ν₁ (ρ.map (fun γ => (γ 0,γ 1))))
{C : ℝ≥0} (hC : 0 < C)
(hcπ : ∀ t : Time, π.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X)
(hcρ : ∀ t : Time, ρ.map (fun γ => γ t) ≤ ENNReal.ofReal C • hausdorff n X)
(hs₀ : ∀ᵐ x ∂ν₀, dist x p ≤ R) (hs₁ : ∀ᵐ x ∂ν₁, dist x p ≤ R) : π=ρ := by
let ξ := (2:ℝ≥0∞)⁻¹ • (π+ρ)
obtain ⟨hprob,hgξ,hpξ,hπξ,hρξ⟩ := half_mixture_plan_data π ρ hgπ hgρ hpπ hpρ
let : IsProbabilityMeasure ξ := hprob
have heπ := optimalPlan_endpoints ν₀ ν₁ π hpπ
have heρ := optimalPlan_endpoints ν₀ ν₁ ρ hpρ
have heξ := optimalPlan_endpoints ν₀ ν₁ ξ hpξ
have hcξ := half_mixture_compression (hausdorff n X) π ρ hcπ hcρ
have hpξ' : IsOptimalCoupling (ξ.map (fun γ => γ 0)) (ξ.map (fun γ => γ 1))
(ξ.map (fun γ => (γ 0,γ 1))) := by rwa [heξ.1,heξ.2]
obtain ⟨g,hg,hgi⟩ := hX.compressed_initial_inverse hn p hR ξ hgξ hpξ' hC hcξ
(by rwa [heξ.1]) (by rwa [heξ.2])
have hhalf : (2:ℝ≥0∞)⁻¹ ≠ 0 := by norm_num
exact measure_eq_of_common_ae_inverse ξ π ρ (measurable_path_eval 0) hg hgi
((Measure.absolutelyContinuous_smul hhalf).trans hπξ.absolutelyContinuous)
((Measure.absolutelyContinuous_smul hhalf).trans hρξ.absolutelyContinuous)
(heπ.1.trans heρ.1.symm)
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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