| import OAI.Geometry.RCD.Transport.CompactCouplingSlack |
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| namespace OAI |
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| /-! |
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| 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] |
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| 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] |
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| 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 |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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