File size: 3,811 Bytes
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 | 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
|