| import Mathlib.MeasureTheory.Measure.Prokhorov | |
| import Mathlib.MeasureTheory.Measure.Support | |
| import Mathlib.MeasureTheory.Integral.Bochner.Set | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! # Compact limits of couplings with vanishing dual slack -/ | |
| open Set Filter MeasureTheory | |
| open scoped Topology | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] | |
| [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace X] [BorelSpace Y] | |
| (μ : Measure X) (ν : Measure Y) | |
| omit [MetricSpace X] [BorelSpace X] [BorelSpace Y] in | |
| theorem coupling_snd_support (π : Measure (X × Y)) (hπ : π.map Prod.snd=ν) | |
| [SecondCountableTopology Y] : ∀ᵐ z ∂π, z.2 ∈ ν.support := by | |
| apply ae_of_ae_map measurable_snd.aemeasurable | |
| rw [hπ] | |
| exact ν.support_mem_ae | |
| theorem compact_coupling_zero_slack [CompactSpace X] [CompactSpace Y] | |
| [SecondCountableTopology X] [SecondCountableTopology Y] | |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] | |
| (P : ℕ → ProbabilityMeasure (X × Y)) | |
| (hl : ∀ k, (P k : Measure (X × Y)).map Prod.fst=μ) | |
| (hr : ∀ k, (P k : Measure (X × Y)).map Prod.snd=ν) | |
| (s : C(X × Y,ℝ)) (hs : ∀ x y, y ∈ ν.support → s (x,y) ≤ 0) | |
| (hlim : Tendsto (fun k => ∫ z, s z ∂(P k : Measure (X × Y))) atTop (𝓝 0)) : | |
| ∃ (Q : ProbabilityMeasure (X × Y)) (ι : ℕ → ℕ), StrictMono ι ∧ | |
| Tendsto (fun k => P (ι k)) atTop (𝓝 Q) ∧ | |
| (Q : Measure (X × Y)).map Prod.fst=μ ∧ | |
| (Q : Measure (X × Y)).map Prod.snd=ν ∧ | |
| ∀ᵐ z ∂(Q : Measure (X × Y)), s z=0 := by | |
| obtain ⟨Q,_,ι,hι,hQ⟩ := isCompact_univ.tendsto_subseq (fun k => mem_univ (P k)) | |
| have hleft : (Q : Measure (X × Y)).map Prod.fst=μ := by | |
| have hh := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous | |
| (fun k => P (ι k)) Q hQ continuous_fst | |
| have he (k : ℕ) : (P (ι k)).map Prod.fst = μ.toProbabilityMeasure := | |
| Subtype.ext (hl (ι k)) | |
| simp only [he] at hh | |
| have heq : μ.toProbabilityMeasure = Q.map Prod.fst := | |
| tendsto_nhds_unique tendsto_const_nhds hh | |
| exact (congrArg (fun r : ProbabilityMeasure X => (r : Measure X)) heq).symm | |
| have hright : (Q : Measure (X × Y)).map Prod.snd=ν := by | |
| have hh := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous | |
| (fun k => P (ι k)) Q hQ continuous_snd | |
| have he (k : ℕ) : (P (ι k)).map Prod.snd = ν.toProbabilityMeasure := | |
| Subtype.ext (hr (ι k)) | |
| simp only [he] at hh | |
| have heq : ν.toProbabilityMeasure = Q.map Prod.snd := | |
| tendsto_nhds_unique tendsto_const_nhds hh | |
| exact (congrArg (fun r : ProbabilityMeasure Y => (r : Measure Y)) heq).symm | |
| have hint : (∫ z, s z ∂(Q : Measure (X × Y)))=0 := | |
| tendsto_nhds_unique ((ProbabilityMeasure.continuous_integral_continuousMap s).tendsto Q |>.comp hQ) | |
| (hlim.comp hι.tendsto_atTop) | |
| have hnonpos : ∀ᵐ z ∂(Q : Measure (X × Y)), s z ≤ 0 := by | |
| filter_upwards [coupling_snd_support (ν := ν) (π := (Q : Measure (X × Y))) hright] with z hz | |
| exact hs z.1 z.2 hz | |
| have hi : Integrable (fun z => -s z) (Q : Measure (X × Y)) := | |
| s.continuous.neg.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) | |
| have he : (fun z => -s z) =ᵐ[(Q : Measure (X × Y))] 0 := | |
| (integral_eq_zero_iff_of_nonneg_ae (by filter_upwards [hnonpos] with z hz; simpa using hz) hi).mp | |
| (by rw [integral_neg,hint,neg_zero]) | |
| refine ⟨Q,ι,hι,hQ,hleft,hright,?_⟩ | |
| filter_upwards [he] with z hz | |
| simpa using hz | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |