| import Mathlib |
| import OAI.Analysis.HardSphere.Bridges.PrimaryObserved |
|
|
| namespace OAI |
|
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| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| open HardSphere.Bridge |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma fiberNormalize_apply_measure {E X : Type*} [MeasurableSpace E] [MeasurableSpace X] |
| (κ : Kernel E X) [IsSFiniteKernel κ] (A : Set (E×X)) (hA : MeasurableSet A) (e : E) : |
| fiberNormalize κ A e=conditionedMeasure (κ e) (Prod.mk e ⁻¹' A) := by |
| change (fiberRestrict _ _ e univ)⁻¹ • fiberRestrict _ _ e=_ |
| rw [fiberRestrict_apply _ hA] |
| simp only [Measure.restrict_apply_univ,conditionedMeasure] |
| rfl |
|
|
| variable {E I : Type*} [MeasurableSpace E] [Fintype I] [DecidableEq I] |
| {X : I→Type*} [∀i,MeasurableSpace (X i)] |
|
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| def ambientFirst (s : Finset I) (σ : Equiv.Perm I) |
| (p : (∀i,X i)×(∀i:s,X i×X (σ.symm i))) : ∀i,X i := |
| Function.updateFinset p.1 s (fun i ↦ (p.2 i).1) |
|
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| def ambientSecond (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (p : (∀i,X i)×(∀i:s,X i×X (σ.symm i))) : ∀i,X i := |
| Function.updateFinset p.1 s (fun i ↦ (reverseSelected σ s hf p.2 i).1) |
|
|
| omit [Fintype I] in |
| lemma ambientFirst_measurable (s : Finset I) (σ : Equiv.Perm I) : |
| Measurable (ambientFirst (X:=X) s σ) := |
| measurable_updateFinset'.comp (measurable_fst.prodMk |
| (Measurable.of_eval (fun i ↦ measurable_fst.comp ((measurable_pi_apply i).comp measurable_snd)))) |
|
|
| omit [Fintype I] in |
| lemma ambientSecond_measurable (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) : |
| Measurable (ambientSecond (X:=X) s σ hf) := |
| measurable_updateFinset'.comp (measurable_fst.prodMk |
| (Measurable.of_eval (fun i ↦ measurable_fst.comp ((measurable_pi_apply i).comp |
| ((reverseSelected σ s hf).measurable.comp measurable_snd))))) |
|
|
| def ambientRefreshEvent (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (A : Set (E×(∀i,X i))) : Set ((E×(∀i,X i))×(∀i:s,X i×X (σ.symm i))) := |
| {p | (p.1.1,ambientFirst s σ (p.1.2,p.2))∈A ∧ |
| (p.1.1,ambientSecond s σ hf (p.1.2,p.2))∈A} |
|
|
| omit [Fintype I] in |
| lemma ambientRefreshEvent_measurable (s : Finset I) (σ : Equiv.Perm I) |
| (hf : ∀i,i∉s→σ i=i) (A : Set (E×(∀i,X i))) (hA : MeasurableSet A) : |
| MeasurableSet (ambientRefreshEvent s σ hf A) := by |
| exact (hA.preimage (measurable_fst.fst.prodMk ((ambientFirst_measurable s σ).comp |
| (measurable_fst.snd.prodMk measurable_snd)))).inter |
| (hA.preimage (measurable_fst.fst.prodMk ((ambientSecond_measurable s σ hf).comp |
| (measurable_fst.snd.prodMk measurable_snd)))) |
|
|
| def ambientRefreshKernel (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) : Kernel (E×(∀i,X i)) (∀i:s,X i×X (σ.symm i)) := |
| fiberNormalize ((finiteKernelPi ρ).comap Prod.fst measurable_fst) (ambientRefreshEvent s σ hf A) |
|
|
| instance ambientRefreshKernel_finite (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) : IsFiniteKernel (ambientRefreshKernel s σ hf ρ A) := by |
| unfold ambientRefreshKernel |
| infer_instance |
|
|
| omit [Fintype I] in |
| lemma ambientRefreshKernel_apply (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) (hA : MeasurableSet A) (e : E) (x : ∀i,X i) : |
| ambientRefreshKernel s σ hf ρ A (e,x)= |
| conditionedMeasure (Measure.pi fun i ↦ ρ i e) |
| {p | (e,ambientFirst s σ (x,p))∈A ∧ (e,ambientSecond s σ hf (x,p))∈A} := by |
| rw [ambientRefreshKernel,fiberNormalize_apply_measure _ _ (ambientRefreshEvent_measurable s σ hf A hA), |
| Kernel.comap_apply,finiteKernelPi_apply] |
| rfl |
|
|
| def ambientSharedKernel (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ξ : Kernel E (∀i,X i)) [IsFiniteKernel ξ] |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) : Kernel E ((∀i,X i)×(∀i:s,X i×X (σ.symm i))) := |
| ξ ⊗ₖ ambientRefreshKernel s σ hf ρ A |
|
|
| instance ambientSharedKernel_finite (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ξ : Kernel E (∀i,X i)) [IsFiniteKernel ξ] |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) : IsFiniteKernel (ambientSharedKernel s σ hf ξ ρ A) := by |
| unfold ambientSharedKernel |
| infer_instance |
|
|
| omit [Fintype I] in |
| lemma ambientSharedKernel_mass (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ξ : Kernel E (∀i,X i)) [IsFiniteKernel ξ] |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) (hA : MeasurableSet A) (e : E) |
| (hp : ∀ᵐx ∂ξ e,0<(Measure.pi fun i ↦ ρ i e) |
| {p | (e,ambientFirst s σ (x,p))∈A ∧ (e,ambientSecond s σ hf (x,p))∈A}) : |
| ambientSharedKernel s σ hf ξ ρ A e univ=ξ e univ := by |
| rw [ambientSharedKernel,Kernel.compProd_apply MeasurableSet.univ] |
| simp only [preimage_univ] |
| calc |
| _ = ∫⁻_x,1 ∂ξ e := by |
| apply lintegral_congr_ae |
| filter_upwards [hp] with x hx |
| apply fiberNormalize_mass _ (ambientRefreshEvent_measurable s σ hf A hA) |
| simpa only [Kernel.comap_apply,finiteKernelPi_apply,ambientRefreshEvent,mem_ofPred_eq] using hx |
| _ = _ := by simp |
|
|
| lemma ambientSharedKernel_support (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ξ : Kernel E (∀i,X i)) [IsFiniteKernel ξ] |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) (hA : MeasurableSet A) (e : E) |
| (B : ∀i:s,Set (X i×X (σ.symm i))) (hB : ∀i,MeasurableSet (B i)) |
| (hρ : ∀i,∀ᵐp ∂ρ i e,p∈B i) : |
| ∀ᵐp ∂ambientSharedKernel s σ hf ξ ρ A e,∀i,p.2 i∈B i := by |
| have hm : ∀ᵐp ∂Measure.pi (fun i ↦ ρ i e),∀i,p i∈B i := by |
| rw [ae_all_iff] |
| exact fun i ↦ (Measure.quasiMeasurePreserving_eval (fun j ↦ ρ j e) i).ae (hρ i) |
| have hS : MeasurableSet {p : (∀i,X i)×(∀i:s,X i×X (σ.symm i)) | ∀i,p.2 i∈B i} := by |
| convert MeasurableSet.iInter (fun i ↦ (hB i).preimage ((measurable_pi_apply i).comp |
| (measurable_snd : Measurable (Prod.snd : (∀i,X i)×(∀i:s,X i×X (σ.symm i))→_)))) using 1 |
| ext p |
| simp only [mem_ofPred_eq,mem_iInter,mem_preimage,Function.comp_apply] |
| apply Kernel.ae_compProd_of_ae_ae hS |
| apply ae_of_all |
| intro x |
| apply (fiberNormalize_ac _ (ambientRefreshEvent_measurable s σ hf A hA) (e,x)).ae_le |
| simp only [Kernel.comap_apply,finiteKernelPi_apply] |
| simp only [Filter.Eventually] at hm ⊢ |
| convert hm using 1 |
|
|
| theorem ambient_common_observation {Y : Type*} [MeasurableSpace Y] [MeasurableEq Y] |
| (s : Finset I) (σ : Equiv.Perm I) (hf : ∀i,i∉s→σ i=i) |
| (ξ : Kernel E (∀i,X i)) [IsFiniteKernel ξ] |
| (ρ : ∀i:s,Kernel E (X i×X (σ.symm i))) [∀i,IsFiniteKernel (ρ i)] |
| (A : Set (E×(∀i,X i))) (hA : MeasurableSet A) (e : E) |
| (mid : ∀i,X i→Y) (hmid : ∀i,Measurable (mid i)) |
| (hp : ∀i:s,∀ᵐp ∂ρ i e,mid i p.1=mid (σ.symm i) p.2) |
| (star a b : I) (hsa : σ star=a) (hbs : σ b=star) : |
| (ambientSharedKernel s σ hf ξ ρ A e).map |
| (primaryFirstObservation σ star a b mid ∘ ambientFirst s σ)= |
| (ambientSharedKernel s σ hf ξ ρ A e).map |
| (primarySecondObservation star b mid ∘ ambientSecond s σ hf) := by |
| apply Measure.map_congr |
| have h := ambientSharedKernel_support s σ hf ξ ρ A hA e |
| (fun i ↦ {p | mid i p.1=mid (σ.symm i) p.2}) |
| (fun i ↦ measurableSet_eq_fun ((hmid i).comp measurable_fst) ((hmid _).comp measurable_snd)) hp |
| filter_upwards [h] with p hp |
| exact primary_observations_eq σ s hf mid p.1 p.2 hp star a b hsa hbs |
|
|
| end HardSphere.Comparison |
| end |
| end |
|
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| end OAI |
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