File size: 3,376 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 | import OAI.Geometry.RCD.Transport.CompactCouplingSlack
namespace OAI
/-! # A saturated dual pair minimizes the compact transport cost -/
open Set Filter MeasureTheory
noncomputable section
namespace RegularPointCoordinates
variable {X Y : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [MeasurableSpace Y] [BorelSpace X] [BorelSpace Y]
[CompactSpace X] [CompactSpace Y]
(μ : Measure X) (ν : Measure Y) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
omit [IsProbabilityMeasure ν] in
theorem coupling_potential_integral (π : Measure (X × Y))
(hl : π.map Prod.fst=μ) (hr : π.map Prod.snd=ν)
(f : C(X,ℝ)) (g : C(Y,ℝ)) (c : ℝ) :
(∫ z, f z.1+g z.2+c ∂π)=(∫ x, f x ∂μ)+(∫ y, g y ∂ν)+c := by
let : IsProbabilityMeasure (π.map Prod.fst) := by rw [hl]; infer_instance
let : IsProbabilityMeasure π := Measure.isProbabilityMeasure_of_map measurable_fst.aemeasurable
have hf : Integrable (fun z : X × Y => f z.1) π :=
(f.continuous.comp continuous_fst).integrable_of_hasCompactSupport
(HasCompactSupport.of_compactSpace _)
have hg : Integrable (fun z : X × Y => g z.2) π :=
(g.continuous.comp continuous_snd).integrable_of_hasCompactSupport
(HasCompactSupport.of_compactSpace _)
rw [integral_add (f := fun z : X × Y => f z.1+g z.2) (g := fun _ : X × Y => c)
(hf.add hg) (integrable_const c),
integral_add (f := fun z : X × Y => f z.1) (g := fun z : X × Y => g z.2) hf hg]
have he : (∫ z, f z.1 ∂π)=∫ x, f x ∂μ := by
rw [← hl,integral_map measurable_fst.aemeasurable f.continuous.aestronglyMeasurable]
have he' : (∫ z, g z.2 ∂π)=∫ y, g y ∂ν := by
rw [← hr,integral_map measurable_snd.aemeasurable g.continuous.aestronglyMeasurable]
simp only [he,he',integral_const,probReal_univ,smul_eq_mul,one_mul]
omit [IsProbabilityMeasure ν] in
theorem compact_dual_cost_minimizer [SecondCountableTopology Y]
(Q : Measure (X × Y)) (hQl : Q.map Prod.fst=μ) (hQr : Q.map Prod.snd=ν)
(f : C(X,ℝ)) (g : C(Y,ℝ)) (cost : C(X × Y,ℝ)) (c : ℝ)
(hdual : ∀ x y, y ∈ ν.support → f x+g y+c ≤ cost (x,y))
(heq : ∀ᵐ z ∂Q, f z.1+g z.2+c=cost z) :
∀ π : Measure (X × Y), π.map Prod.fst=μ → π.map Prod.snd=ν →
(∫ z, cost z ∂Q) ≤ ∫ z, cost z ∂π := by
intro π hl hr
let : IsProbabilityMeasure (π.map Prod.fst) := by rw [hl]; infer_instance
let : IsProbabilityMeasure π := Measure.isProbabilityMeasure_of_map measurable_fst.aemeasurable
have hle := integral_mono_ae
(((f.continuous.comp continuous_fst).add (g.continuous.comp continuous_snd)).add continuous_const
|>.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
(cost.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
(by
filter_upwards [coupling_snd_support (ν := ν) (π := π) hr] with z hz
exact hdual z.1 z.2 hz)
calc
(∫ z, cost z ∂Q) = ∫ z, f z.1+g z.2+c ∂Q := integral_congr_ae (by
filter_upwards [heq] with z hz
exact hz.symm)
_ = (∫ x, f x ∂μ)+(∫ y, g y ∂ν)+c := coupling_potential_integral μ ν Q hQl hQr f g c
_ = (∫ z, f z.1+g z.2+c ∂π) := (coupling_potential_integral μ ν π hl hr f g c).symm
_ ≤ (∫ z, cost z ∂π) := hle
end RegularPointCoordinates
end
end OAI
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