File size: 8,881 Bytes
67aee1d | 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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {E X : Type*} [MeasurableSpace E] [MeasurableSpace X]
def parameterSuccessKernel (κ : Kernel E X) [IsFiniteKernel κ] (S : Set (E×X))
(hS : MeasurableSet S) : Kernel E X where
toFun e := (κ e (Prod.mk e ⁻¹' S))⁻¹ • (κ e).restrict (Prod.mk e ⁻¹' S)
measurable' := by
refine Measure.measurable_of_measurable_coe _ fun B hB ↦ ?_
simp only [Measure.smul_apply,Measure.restrict_apply hB,smul_eq_mul]
have hm := Kernel.measurable_kernel_prodMk_left (κ:=κ) hS
have hi := Kernel.measurable_kernel_prodMk_left (κ:=κ)
((MeasurableSet.univ.prod hB).inter hS)
apply hm.inv.mul
convert hi using 1
ext e
congr 1
ext x
simp
lemma parameterSuccessKernel_apply (κ : Kernel E X) [IsFiniteKernel κ] (S : Set (E×X))
(hS : MeasurableSet S) (e : E) :
parameterSuccessKernel κ S hS e=
(κ e (Prod.mk e ⁻¹' S))⁻¹ • (κ e).restrict (Prod.mk e ⁻¹' S) := rfl
lemma parameterSuccessKernel_mass_le (κ : Kernel E X) [IsFiniteKernel κ] (S : Set (E×X))
(hS : MeasurableSet S) (e : E) :
parameterSuccessKernel κ S hS e univ ≤ 1 := by
simp only [parameterSuccessKernel_apply,Measure.smul_apply,
Measure.restrict_apply_univ,smul_eq_mul]
exact ENNReal.inv_mul_le_one _
instance parameterSuccessKernel_finite (κ : Kernel E X) [IsFiniteKernel κ] (S : Set (E×X))
(hS : MeasurableSet S) : IsFiniteKernel (parameterSuccessKernel κ S hS) :=
⟨1,by simp,parameterSuccessKernel_mass_le κ S hS⟩
lemma parameterSuccessKernel_mass (κ : Kernel E X) [IsFiniteKernel κ]
(S : Set (E×X)) (hS : MeasurableSet S) {e : E}
(h : 0 < κ e (Prod.mk e ⁻¹' S)) :
parameterSuccessKernel κ S hS e univ=1 := by
simp only [parameterSuccessKernel_apply,Measure.smul_apply,
Measure.restrict_apply_univ,smul_eq_mul]
exact ENNReal.inv_mul_cancel h.ne' (measure_ne_top _ _)
lemma parameterSuccessKernel_markov (κ : Kernel E X) [IsFiniteKernel κ]
(S : Set (E×X)) (hS : MeasurableSet S)
(h : ∀e,0 < κ e (Prod.mk e ⁻¹' S)) :
IsMarkovKernel (parameterSuccessKernel κ S hS) :=
⟨fun e ↦ ⟨parameterSuccessKernel_mass κ S hS (h e)⟩⟩
lemma parameterSuccessKernel_le (κ : Kernel E X) [IsFiniteKernel κ] (S : Set (E×X))
(hS : MeasurableSet S) {e : E} {c : ℝ≥0∞}
(hc : c ≤ κ e (Prod.mk e ⁻¹' S)) :
parameterSuccessKernel κ S hS e ≤ c⁻¹ • κ e := by
rw [parameterSuccessKernel_apply]
intro B
simp only [Measure.smul_apply,smul_eq_mul]
exact mul_le_mul' (ENNReal.inv_le_inv.mpr hc) (Measure.restrict_le_self B)
lemma parameterSuccessKernel_ac (κ : Kernel E X) [IsFiniteKernel κ] (S : Set (E×X))
(hS : MeasurableSet S) (e : E) :
parameterSuccessKernel κ S hS e ≪ κ e := by
rw [parameterSuccessKernel_apply]
exact Measure.smul_absolutelyContinuous.trans (Measure.restrict_le_self.absolutelyContinuous)
lemma parameterSuccessKernel_support (κ : Kernel E X) [IsFiniteKernel κ] (S : Set (E×X))
(hS : MeasurableSet S) (e : E) :
∀ᵐx ∂parameterSuccessKernel κ S hS e,(e,x)∈S := by
have hs : MeasurableSet (Prod.mk e ⁻¹' S) := hS.preimage measurable_prodMk_left
rw [ae_iff]
change parameterSuccessKernel κ S hS e (Prod.mk e ⁻¹' S)ᶜ=0
simp [parameterSuccessKernel_apply,Measure.restrict_apply hs.compl]
end HardSphere.Comparison
namespace HardSphere.Comparison
lemma exists_bounded_auxiliary_kernel {X Z : Type*} [MeasurableSpace X] [MeasurableSpace Z]
[StandardBorelSpace Z] [Nonempty Z] (P : Measure X) [IsProbabilityMeasure P]
(ρ : Measure (X×Z)) [IsProbabilityMeasure ρ] {G : ℝ≥0∞} (hG : G≠⊤)
(hρ : ρ.fst≤G • P) :
∃ κ : Kernel X Z,IsFiniteKernel κ ∧ P.compProd κ=ρ ∧
(∀x,κ x univ≤G) ∧ (∫⁻x,κ x univ ∂P)=1 := by
have hac : ρ.fst ≪ P := (Measure.absolutelyContinuous_of_le hρ).trans Measure.smul_absolutelyContinuous
have hbd : ∀ᵐx ∂P,ρ.fst.rnDeriv P x≤G := by
apply ae_le_of_forall_setLIntegral_le_of_sigmaFinite (ρ.fst.measurable_rnDeriv P)
intro S hS hSf
simpa only [lintegral_const,Measure.restrict_apply_univ,smul_eq_mul,Measure.smul_apply]
using (Measure.setLIntegral_rnDeriv_le (μ:=ρ.fst) (ν:=P) S).trans (hρ S)
let f : X→ℝ≥0∞ := fun x ↦ min (ρ.fst.rnDeriv P x) G
have hfm : Measurable f := (ρ.fst.measurable_rnDeriv P).min measurable_const
have hfe : f=ᵐ[P]ρ.fst.rnDeriv P := hbd.mono (fun x hx ↦ min_eq_left hx)
have hw : P.withDensity f=ρ.fst := (withDensity_congr_ae hfe).trans (Measure.withDensity_rnDeriv_eq _ _ hac)
let κ := ρ.condKernel.withDensity (fun x _ ↦ f x)
have hκ : IsFiniteKernel κ := Kernel.isFiniteKernel_withDensity_of_bounded _ hG (fun x _ ↦ min_le_right _ _)
let _ := hκ
have hκeq (x : X) : κ x=f x • ρ.condKernel x := by
change ρ.condKernel.withDensity (fun (x : X) (_ : Z) ↦ f x) x = _
rw [Kernel.withDensity_apply _ (show Measurable (Function.uncurry (fun (x : X) (_ : Z) ↦ f x)) from hfm.comp measurable_fst)]
exact withDensity_const _
have hmass (x : X) : κ x univ=f x := by rw [hκeq]; simp
have hjoint : P.compProd κ=ρ := by
rw [←ρ.disintegrate ρ.condKernel,←hw]
ext S hS
rw [Measure.compProd_apply hS,Measure.compProd_apply hS,
lintegral_withDensity_eq_lintegral_mul _ hfm (Kernel.measurable_kernel_prodMk_left hS)]
apply lintegral_congr
intro x
rw [hκeq,Measure.smul_apply,smul_eq_mul]
rfl
refine ⟨κ,hκ,hjoint,fun x ↦ ?_,?_⟩
· rw [hmass]; exact min_le_right _ _
· have hu := congrArg (fun μ : Measure (X×Z) ↦ μ univ) hjoint
simpa only [Measure.compProd_apply MeasurableSet.univ,preimage_univ,measure_univ] using hu
lemma exists_parameter_auxiliary {E X Z : Type*} [MeasurableSpace E]
[MeasurableSpace X] [MeasurableSpace Z]
[MeasurableSpace.CountableOrCountablyGenerated E X]
[StandardBorelSpace Z] [Nonempty Z]
(P : Kernel E X) [IsMarkovKernel P]
(ρ : Kernel E (X×Z)) [IsMarkovKernel ρ]
{G : ℝ≥0∞} (hG : G ≠ ⊤) (hρ : ∀e,ρ.fst e ≤ G • P e) :
∃ κ : Kernel (E×X) Z, IsFiniteKernel κ ∧ P.compProd κ=ρ ∧
(∀e x,κ (e,x) univ ≤ G) ∧ (∀e,∫⁻x,κ (e,x) univ ∂P e=1) := by
have hac (e : E) : ρ.fst e ≪ P e :=
(Measure.absolutelyContinuous_of_le (hρ e)).trans Measure.smul_absolutelyContinuous
have hbd (e : E) : ∀ᵐx ∂P e,ρ.fst.rnDeriv P e x ≤ G := by
apply ae_le_of_forall_setLIntegral_le_of_sigmaFinite (ρ.fst.measurable_rnDeriv_right P e)
intro S hS hSf
simpa only [lintegral_const,Measure.restrict_apply_univ,smul_eq_mul,Measure.smul_apply]
using (Kernel.setLIntegral_rnDeriv_le (κ:=ρ.fst) (η:=P) (a:=e) hS).trans (hρ e S)
let f : E×X → ℝ≥0∞ := fun p ↦ min (ρ.fst.rnDeriv P p.1 p.2) G
have hfm : Measurable f := (ρ.fst.measurable_rnDeriv P).min measurable_const
have hw (e : E) : (P e).withDensity (fun x ↦ f (e,x))=ρ.fst e := by
have hfe : (fun x ↦ f (e,x))=ᵐ[P e]ρ.fst.rnDeriv P e :=
(hbd e).mono (fun x hx ↦ min_eq_left hx)
rw [withDensity_congr_ae hfe]
rw [←Kernel.withDensity_apply P (ρ.fst.measurable_rnDeriv P)]
exact Kernel.withDensity_rnDeriv_eq (hac e)
let κ := ρ.condKernel.withDensity (fun p (_ : Z) ↦ f p)
have hκ : IsFiniteKernel κ := Kernel.isFiniteKernel_withDensity_of_bounded _ hG
(fun p _ ↦ min_le_right _ _)
let _ := hκ
have hκeq (p : E×X) : κ p=f p • ρ.condKernel p := by
change ρ.condKernel.withDensity (fun (p : E×X) (_ : Z) ↦ f p) p=_
rw [Kernel.withDensity_apply _ (show Measurable (Function.uncurry (fun (p : E×X) (_ : Z) ↦ f p)) from hfm.comp measurable_fst)]
exact withDensity_const _
have hmass (p : E×X) : κ p univ=f p := by rw [hκeq];simp
have hjoint : P.compProd κ=ρ := by
rw [←ρ.disintegrate ρ.condKernel]
apply Kernel.ext
intro e
ext S hS
rw [Kernel.compProd_apply hS,Kernel.compProd_apply hS,←hw e]
have hg : Measurable (fun x : X ↦ ρ.condKernel (e,x) (Prod.mk x ⁻¹' S)) :=
Kernel.measurable_kernel_prodMk_left (κ:=ρ.condKernel.comap (fun x : X ↦ (e,x)) measurable_prodMk_left) hS
rw [lintegral_withDensity_eq_lintegral_mul _
(show Measurable (fun x : X ↦ f (e,x)) from hfm.comp measurable_prodMk_left) hg]
apply lintegral_congr
intro x
rw [hκeq,Measure.smul_apply,smul_eq_mul]
rfl
refine ⟨κ,hκ,hjoint,fun e x ↦ ?_,fun e ↦ ?_⟩
· rw [hmass];exact min_le_right _ _
· have hu := congrArg (fun η : Kernel E (X×Z) ↦ η e univ) hjoint
simpa only [Kernel.compProd_apply MeasurableSet.univ,preimage_univ,measure_univ] using hu
end HardSphere.Comparison
end
end OAI
|