File size: 3,580 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 | 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
|