| 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 |
| |
