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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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Geometry.RCD.Transport.PairedPathUniqueness
import OAI.Geometry.RCD.Transport.PairedSuffixUniqueness
namespace OAI
/-! # Explicit optimal half mixtures with their compression and domination -/
open Set Filter MeasureTheory TopologicalSpace
open scoped ENNReal
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [SeparableSpace X]
attribute [local instance] pathBorel pathBorelSpace
omit [SeparableSpace X] in
theorem half_mixture_plan_data {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)))) :
let τ := (2:ℝ≥0∞)⁻¹ • (π+ρ)
IsProbabilityMeasure τ ∧ (∀ᵐ γ ∂τ, IsGeodesic γ) ∧
IsOptimalCoupling ν₀ ν₁ (τ.map (fun γ => (γ 0,γ 1))) ∧
(2:ℝ≥0∞)⁻¹ • π ≤ τ ∧ (2:ℝ≥0∞)⁻¹ • ρ ≤ τ := by
let c : ℝ≥0∞ := (2:ℝ≥0∞)⁻¹
let τ := c • (π+ρ)
have hcc : c+c=1 := by
dsimp only [c]
rw [←two_mul,ENNReal.mul_inv_cancel (by norm_num) (by norm_num)]
have hprob : IsProbabilityMeasure τ := ⟨by
change c*(π univ+ρ univ)=1
simp only [measure_univ,mul_add,mul_one,hcc]⟩
have hm := (measurable_path_eval (X:=X) 0).prodMk (measurable_path_eval 1)
have he (ν : Measure X) : c • (ν+ν)=ν := half_smul_self ν
have hopt : IsOptimalCoupling ν₀ ν₁ (τ.map (fun γ => (γ 0,γ 1))) := by
rw [show τ=c • (π+ρ) from rfl,Measure.map_smul _ hm.aemeasurable,Measure.map_add _ _ hm]
refine ⟨?_,?_,?_⟩
· rw [Measure.map_smul _ measurable_fst.aemeasurable,Measure.map_add _ _ measurable_fst,
hpπ.1,hpρ.1,he]
· rw [Measure.map_smul _ measurable_snd.aemeasurable,Measure.map_add _ _ measurable_snd,
hpπ.2.1,hpρ.2.1,he]
· intro η hη₀ hη₁
rw [transportCost_smul,transportCost_add,mul_add]
calc
_ ≤ c*transportCost η+c*transportCost η :=
add_le_add (mul_le_mul' le_rfl (hpπ.2.2 η hη₀ hη₁))
(mul_le_mul' le_rfl (hpρ.2.2 η hη₀ hη₁))
_ = _ := by rw [←add_mul,hcc,one_mul]
refine ⟨hprob,?_,hopt,?_,?_⟩
· exact ((ae_add_measure_iff.mpr ⟨hgπ,hgρ⟩).filter_mono Measure.smul_absolutelyContinuous.ae_le)
· exact smul_le_smul_left c (Measure.le_add_right le_rfl)
· exact smul_le_smul_left c (Measure.le_add_left le_rfl)
omit [SeparableSpace X] in
theorem half_mixture_compression (μ : Measure X) (π ρ : Measure C(Time,X))
{C : ℝ≥0∞} (hπ : ∀ t : Time, π.map (fun γ => γ t) ≤ C • μ)
(hρ : ∀ t : Time, ρ.map (fun γ => γ t) ≤ C • μ) (t : Time) :
((2:ℝ≥0∞)⁻¹ • (π+ρ)).map (fun γ => γ t) ≤ C • μ := by
rw [half_mixture_map_eval]
exact (_root_.smul_le_smul_left ((2:ℝ≥0∞)⁻¹) (add_le_add (hπ t) (hρ t))).trans_eq (half_smul_self _)
end RegularPointCoordinates
end
end OAI
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