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dd14c26 | 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 78 79 80 81 82 83 84 85 86 | import Mathlib
import OAI.Probability.LatticeSurface.Dynamics.CompactControls
namespace OAI
open Set MeasureTheory
namespace CompactCoupling
variable {X Y U : Type*}
[TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [T2Space X] [CompactSpace X] [SecondCountableTopology X]
[TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [T2Space Y] [CompactSpace Y] [SecondCountableTopology Y]
[TopologicalSpace U] [MeasurableSpace U] [BorelSpace U] [T2Space U] [CompactSpace U] [SecondCountableTopology U]
noncomputable def family (μ : ProbabilityMeasure X) (ν : ProbabilityMeasure Y) : Set (ProbabilityMeasure (X×Y)) :=
{π | π.map Prod.fst=μ ∧ π.map Prod.snd=ν}
lemma compact_family (μ : ProbabilityMeasure X) (ν : ProbabilityMeasure Y) : IsCompact (family μ ν) := by
exact ((isClosed_eq (ProbabilityMeasure.continuous_map continuous_fst) continuous_const).inter
(isClosed_eq (ProbabilityMeasure.continuous_map continuous_snd) continuous_const)).isCompact
omit [TopologicalSpace X] [BorelSpace X] [T2Space X] [CompactSpace X]
[SecondCountableTopology X] [BorelSpace Y] [T2Space Y] [CompactSpace Y] in
lemma nonempty_family (μ : ProbabilityMeasure X) (ν : ProbabilityMeasure Y) : (family μ ν).Nonempty := by
refine ⟨⟨(μ:Measure X).prod (ν:Measure Y),inferInstance⟩,?_,?_⟩
· apply Subtype.ext
change Measure.map Prod.fst ((μ:Measure X).prod (ν:Measure Y))=(μ:Measure X)
simp
· apply Subtype.ext
change Measure.map Prod.snd ((μ:Measure X).prod (ν:Measure Y))=(ν:Measure Y)
simp
omit [T2Space X] [CompactSpace X] [T2Space Y] [CompactSpace Y]
[T2Space U] [CompactSpace U] in
lemma step_stays (μ : ProbabilityMeasure X) (ν : ProbabilityMeasure Y)
(ρ : Measure U) [IsProbabilityMeasure ρ]
(T : X×U → X) (S : Y×U → Y) (hT : Continuous T) (hS : Continuous S)
(hμ : Measure.map T ((μ:Measure X).prod ρ)=μ)
(hν : Measure.map S ((ν:Measure Y).prod ρ)=ν)
(π : ProbabilityMeasure (X×Y)) (hπ : π∈family μ ν) :
CompactControls.step ρ (fun z : (X×Y)×U => (T (z.1.1,z.2),S (z.1.2,z.2)))
((hT.comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd)).prodMk
(hS.comp ((continuous_snd.comp continuous_fst).prodMk continuous_snd))) π∈family μ ν := by
have hp : Measure.map Prod.fst (π:Measure (X×Y))=μ := congrArg Subtype.val hπ.1
have hq : Measure.map Prod.snd (π:Measure (X×Y))=ν := congrArg Subtype.val hπ.2
have hs : Measurable (fun z : (X×Y)×U => (T (z.1.1,z.2),S (z.1.2,z.2))) := ((hT.comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd)).prodMk
(hS.comp ((continuous_snd.comp continuous_fst).prodMk continuous_snd))).measurable
constructor
· apply Subtype.ext
change Measure.map Prod.fst (Measure.map _ ((π:Measure (X×Y)).prod ρ))=(μ:Measure X)
rw [Measure.map_map measurable_fst hs]
change Measure.map (T ∘ Prod.map Prod.fst id) ((π:Measure (X×Y)).prod ρ)=(μ:Measure X)
rw [← Measure.map_map hT.measurable (measurable_fst.prodMap measurable_id),
← Measure.map_prod_map _ _ measurable_fst measurable_id,Measure.map_id,hp]
exact hμ
· apply Subtype.ext
change Measure.map Prod.snd (Measure.map _ ((π:Measure (X×Y)).prod ρ))=(ν:Measure Y)
rw [Measure.map_map measurable_snd hs]
change Measure.map (S ∘ Prod.map Prod.snd id) ((π:Measure (X×Y)).prod ρ)=(ν:Measure Y)
rw [← Measure.map_map hS.measurable (measurable_snd.prodMap measurable_id),
← Measure.map_prod_map _ _ measurable_snd measurable_id,Measure.map_id,hq]
exact hν
omit [T2Space U] in
theorem exists_zero {I : Type*} (μ : ProbabilityMeasure X) (ν : ProbabilityMeasure Y)
(ρ : Measure U) [IsProbabilityMeasure ρ] (hρ : ∀ u,u∈ρ.support) (u₀ : U)
(T : I → X×U → X) (S : I → Y×U → Y) (hT : ∀ i,Continuous (T i)) (hS : ∀ i,Continuous (S i))
(hμ : ∀ i,Measure.map (T i) ((μ:Measure X).prod ρ)=μ)
(hν : ∀ i,Measure.map (S i) ((ν:Measure Y).prod ρ)=ν)
(d H : X×Y → ℝ) (hd : Continuous d) (hH : Continuous H)
(hdec : ∀ i z u,d (T i (z.1,u),S i (z.2,u))≤d z)
(hdown : ∀ i z,H (T i (z.1,u₀),S i (z.2,u₀))≤H z)
(hfixed : ∀ z,(∀ i,H (T i (z.1,u₀),S i (z.2,u₀))=H z) → d z=0) :
∃ π : ProbabilityMeasure (X×Y),π∈family μ ν ∧ ∀ᵐ z∂(π:Measure (X×Y)),d z=0 := by
exact CompactControls.exists_zero ρ hρ u₀
(fun i z => (T i (z.1.1,z.2),S i (z.1.2,z.2)))
(fun i => ((hT i).comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd)).prodMk
((hS i).comp ((continuous_snd.comp continuous_fst).prodMk continuous_snd)))
(family μ ν) (compact_family μ ν) (nonempty_family μ ν)
(fun π hπ i => step_stays μ ν ρ (T i) (S i) (hT i) (hS i) (hμ i) (hν i) π hπ)
d H hd hH hdec hdown hfixed
end CompactCoupling
end OAI
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