File size: 3,914 Bytes
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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import OAI.Geometry.RCD.Transport.CompressedInitialInverse
namespace OAI
/-! # Measurable inverses for every evaluation, including both endpoints -/
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
end
end OAI
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