| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.FiberDisintegrate |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| variable {E I : Type*} [MeasurableSpace E] [Fintype I] |
| variable {X : I→Type*} [∀i,MeasurableSpace (X i)] |
|
|
| def blockGlue (p : I→Prop) [DecidablePred p] |
| (z : (E×(∀i : {i // ¬p i},X i))×(∀i : {i // p i},X i)) : E×(∀i,X i) := |
| (z.1.1,(MeasurableEquiv.piEquivPiSubtypeProd X p).symm (z.2,z.1.2)) |
|
|
| lemma blockGlue_preserving (μ : Measure E) [SFinite μ] (ν : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (ν i)] |
| (p : I→Prop) [DecidablePred p] : |
| MeasurePreserving (blockGlue (E:=E) p) |
| ((μ.prod (Measure.pi fun i : {i // ¬p i} ↦ ν i)).prod (Measure.pi fun i : {i // p i} ↦ ν i)) |
| (μ.prod (Measure.pi ν)) := by |
| have h₁ := measurePreserving_prodAssoc μ (Measure.pi fun i : {i // ¬p i} ↦ ν i) (Measure.pi fun i : {i // p i} ↦ ν i) |
| have h₂ := (MeasurePreserving.id μ).prod (Measure.measurePreserving_swap |
| (μ:=Measure.pi fun i : {i // ¬p i} ↦ ν i) (ν:=Measure.pi fun i : {i // p i} ↦ ν i)) |
| have h₃ := (MeasurePreserving.id μ).prod ((measurePreserving_piEquivPiSubtypeProd ν p).symm (MeasurableEquiv.piEquivPiSubtypeProd X p)) |
| exact h₃.comp (h₂.comp h₁) |
|
|
| omit [Fintype I] in |
| lemma blockGlue_measurable (p : I→Prop) [DecidablePred p] : Measurable (blockGlue (E:=E) (X:=X) p) := |
| measurable_fst.fst.prodMk ((MeasurableEquiv.piEquivPiSubtypeProd X p).symm.measurable.comp (measurable_snd.prodMk measurable_fst.snd)) |
|
|
| def blockEvent (p : I→Prop) [DecidablePred p] (B : Set (E×(∀i,X i))) := blockGlue p ⁻¹' B |
|
|
| def exposedLaw (μ : Measure E) [SFinite μ] (ν : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (ν i)] |
| (p : I→Prop) [DecidablePred p] (B : Set (E×(∀i,X i))) : Measure (E×(∀i : {i // ¬p i},X i)) := |
| (μ.prod (Measure.pi fun i : {i // ¬p i} ↦ ν i)).withDensity |
| (fun w ↦ fiberRestrict (Kernel.const _ (Measure.pi fun i : {i // p i} ↦ ν i)) (blockEvent p B) w univ) |
|
|
| lemma exposed_block_reconstruct (μ : Measure E) [SFinite μ] (ν : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (ν i)] |
| (p : I→Prop) [DecidablePred p] {B : Set (E×(∀i,X i))} (hB : MeasurableSet B) |
| [SFinite (exposedLaw μ ν p B)] : |
| ((exposedLaw μ ν p B).compProd |
| (fiberNormalize (Kernel.const _ (Measure.pi fun i : {i // p i} ↦ ν i)) (blockEvent p B))).map (blockGlue p)= |
| (μ.prod (Measure.pi ν)).restrict B := by |
| have hC : MeasurableSet (blockEvent p B) := hB.preimage (blockGlue_measurable p) |
| have h := fiber_disintegrate (μ.prod (Measure.pi fun i : {i // ¬p i} ↦ ν i)) |
| (Kernel.const _ (Measure.pi fun i : {i // p i} ↦ ν i)) hC |
| change ((exposedLaw μ ν p B).compProd _).map _=_ |
| rw [show (exposedLaw μ ν p B).compProd (fiberNormalize (Kernel.const _ (Measure.pi fun i : {i // p i} ↦ ν i)) (blockEvent p B))= |
| ((μ.prod (Measure.pi fun i : {i // ¬p i} ↦ ν i)).prod (Measure.pi fun i : {i // p i} ↦ ν i)).restrict (blockEvent p B) by simpa only [Measure.compProd_const,exposedLaw] using h] |
| exact ((blockGlue_preserving μ ν p).restrict_preimage hB).map_eq |
|
|
| lemma exposed_block_positive (μ : Measure E) [SFinite μ] (ν : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (ν i)] |
| (p : I→Prop) [DecidablePred p] {B : Set (E×(∀i,X i))} (hB : MeasurableSet B) : |
| ∀ᵐw ∂exposedLaw μ ν p B,0<(Measure.pi fun i : {i // p i} ↦ ν i) {v | blockGlue p (w,v)∈B} := by |
| exact exposed_mass_positive (μ.prod (Measure.pi fun i : {i // ¬p i} ↦ ν i)) |
| (Kernel.const _ (Measure.pi fun i : {i // p i} ↦ ν i)) (hB.preimage (blockGlue_measurable p)) |
|
|
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
