| 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 |
| |
