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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 | import Mathlib
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {U X Y : Type*} [MeasurableSpace U] [MeasurableSpace X] [MeasurableSpace Y]
def successKernel (κ : Kernel U X) (S : Set X) (hS : MeasurableSet S) : Kernel U X where
toFun u := (κ u S)⁻¹ • (κ u).restrict S
measurable' := by
refine Measure.measurable_of_measurable_coe _ fun B hB ↦ ?_
simp only [Measure.smul_apply,Measure.restrict_apply hB,smul_eq_mul]
exact (κ.measurable_coe hS).inv.mul (κ.measurable_coe (hB.inter hS))
lemma successKernel_apply (κ : Kernel U X) (S : Set X) (hS : MeasurableSet S) (u : U) :
successKernel κ S hS u=(κ u S)⁻¹ • (κ u).restrict S := rfl
lemma successKernel_mass_le (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S) (u : U) :
successKernel κ S hS u univ ≤ 1 := by
simp only [successKernel_apply,Measure.smul_apply,Measure.restrict_apply_univ,smul_eq_mul]
exact ENNReal.inv_mul_le_one _
instance successKernel_finite (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S) :
IsFiniteKernel (successKernel κ S hS) := ⟨1,by simp,successKernel_mass_le κ S hS⟩
lemma successKernel_mass (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S)
{u : U} (h : 0<κ u S) : successKernel κ S hS u univ=1 := by
simp only [successKernel_apply,Measure.smul_apply,Measure.restrict_apply_univ,smul_eq_mul]
exact ENNReal.inv_mul_cancel h.ne' (measure_ne_top _ _)
lemma successKernel_le (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S)
{u : U} {c : ℝ≥0∞} (hc : c ≤ κ u S) :
successKernel κ S hS u ≤ c⁻¹ • κ u := by
rw [successKernel_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 successKernel_support (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S) (u : U) :
∀ᵐ x ∂successKernel κ S hS u,x∈S := by
rw [ae_iff]
change successKernel κ S hS u Sᶜ=0
simp [successKernel_apply,Measure.restrict_apply hS.compl]
def pairedFactor (μ : Measure U) (κ : Kernel U X) (η : Kernel U Y)
(S : Set X) (T : Set Y) (hS : MeasurableSet S) (hT : MeasurableSet T)
[IsFiniteKernel κ] [IsFiniteKernel η] : Measure (U×(X×Y)) :=
μ.compProd ((successKernel κ S hS).prod (successKernel η T hT))
lemma pairedFactor_mass (μ : Measure U) [IsProbabilityMeasure μ]
(κ : Kernel U X) (η : Kernel U Y) [IsFiniteKernel κ] [IsFiniteKernel η]
(S : Set X) (T : Set Y) (hS : MeasurableSet S) (hT : MeasurableSet T)
(hκ : ∀ᵐ u ∂μ,0<κ u S) (hη : ∀ᵐ u ∂μ,0<η u T) :
pairedFactor μ κ η S T hS hT univ=1 := by
rw [pairedFactor,Measure.compProd_apply MeasurableSet.univ]
have hea : (fun u ↦ ((successKernel κ S hS).prod (successKernel η T hT)) u
((fun p : X×Y ↦ (u,p))⁻¹' univ))=ᵐ[μ] fun _ ↦ 1 := by
filter_upwards [hκ,hη] with u hu hv
simp only [preimage_univ,Kernel.prod_apply]
rw [show (Set.univ : Set (X×Y))=(Set.univ:Set X)×ˢ(Set.univ:Set Y) from Set.univ_prod_univ.symm,
Measure.prod_prod,successKernel_mass κ S hS hu,successKernel_mass η T hT hv,one_mul]
rw [lintegral_congr_ae hea]
simp
lemma pairedFactor_fst (μ : Measure U) [SFinite μ]
(κ : Kernel U X) (η : Kernel U Y) [IsFiniteKernel κ] [IsFiniteKernel η]
(S : Set X) (T : Set Y) (hS : MeasurableSet S) (hT : MeasurableSet T)
(hη : ∀ᵐ u ∂μ,0<η u T) :
(pairedFactor μ κ η S T hS hT).map (fun p ↦ (p.1,p.2.1))=
μ.compProd (successKernel κ S hS) := by
have hm : Measurable (fun p : U×(X×Y) ↦ (p.1,p.2.1)) := by fun_prop
ext B hB
rw [Measure.map_apply hm hB,
pairedFactor,Measure.compProd_apply (hB.preimage hm),
Measure.compProd_apply hB]
apply lintegral_congr_ae
filter_upwards [hη] with u hu
rw [Kernel.prod_apply]
have he : (fun p : X×Y ↦ (u,p))⁻¹' ((fun p : U×(X×Y) ↦ (p.1,p.2.1))⁻¹' B)=
((fun x : X ↦ (u,x))⁻¹' B)×ˢ(univ : Set Y) := by ext p; simp
rw [he,Measure.prod_prod,successKernel_mass η T hT hu,mul_one]
lemma pairedFactor_snd (μ : Measure U) [SFinite μ]
(κ : Kernel U X) (η : Kernel U Y) [IsFiniteKernel κ] [IsFiniteKernel η]
(S : Set X) (T : Set Y) (hS : MeasurableSet S) (hT : MeasurableSet T)
(hκ : ∀ᵐ u ∂μ,0<κ u S) :
(pairedFactor μ κ η S T hS hT).map (fun p ↦ (p.1,p.2.2))=
μ.compProd (successKernel η T hT) := by
have hm : Measurable (fun p : U×(X×Y) ↦ (p.1,p.2.2)) := by fun_prop
ext B hB
rw [Measure.map_apply hm hB,
pairedFactor,Measure.compProd_apply (hB.preimage hm),
Measure.compProd_apply hB]
apply lintegral_congr_ae
filter_upwards [hκ] with u hu
rw [Kernel.prod_apply]
have he : (fun p : X×Y ↦ (u,p))⁻¹' ((fun p : U×(X×Y) ↦ (p.1,p.2.2))⁻¹' B)=
(univ : Set X)×ˢ((fun y : Y ↦ (u,y))⁻¹' B) := by ext p; simp
rw [he,Measure.prod_prod,successKernel_mass κ S hS hu,one_mul]
lemma pairedFactor_support (μ : Measure U) [SFinite μ]
(κ : Kernel U X) (η : Kernel U Y) [IsFiniteKernel κ] [IsFiniteKernel η]
(S : Set X) (T : Set Y) (hS : MeasurableSet S) (hT : MeasurableSet T) :
∀ᵐ p ∂pairedFactor μ κ η S T hS hT,p.2.1∈S ∧ p.2.2∈T := by
rw [pairedFactor,Measure.ae_compProd_iff (by measurability)]
exact ae_of_all _ (fun u ↦ by
rw [Kernel.prod_apply]
apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr
filter_upwards [successKernel_support κ S hS u] with x hx
filter_upwards [successKernel_support η T hT u] with y hy
exact ⟨hx,hy⟩)
lemma successKernel_absolutelyContinuous (κ : Kernel U X) (S : Set X)
(hS : MeasurableSet S) (u : U) : successKernel κ S hS u ≪ κ u := by
rw [successKernel_apply]
exact Measure.smul_absolutelyContinuous.trans (Measure.restrict_le_self.absolutelyContinuous)
lemma pairedFactor_midpoint {E : Type*} [MeasurableSpace E] [MeasurableEq E]
(μ : Measure U) [SFinite μ] (κ : Kernel U X) (η : Kernel U Y)
[IsFiniteKernel κ] [IsFiniteKernel η] (π : X→E) (σ : Y→E) (uval : U→E)
(hπ : Measurable π) (hσ : Measurable σ) (huval : Measurable uval)
(S : Set X) (T : Set Y) (hS : MeasurableSet S) (hT : MeasurableSet T)
(hκ : ∀ᵐ u ∂μ,∀ᵐ x ∂κ u,π x=uval u)
(hη : ∀ᵐ u ∂μ,∀ᵐ y ∂η u,σ y=uval u) :
∀ᵐ p ∂pairedFactor μ κ η S T hS hT,π p.2.1=uval p.1 ∧ σ p.2.2=uval p.1 := by
rw [pairedFactor,Measure.ae_compProd_iff (by measurability)]
filter_upwards [hκ,hη] with u hu hv
rw [Kernel.prod_apply]
apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr
filter_upwards [(successKernel_absolutelyContinuous κ S hS u).ae_le hu] with x hx
filter_upwards [(successKernel_absolutelyContinuous η T hT u).ae_le hv] with y hy
exact ⟨hx,hy⟩
end HardSphere.Comparison
end
end
end
end OAI
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