File size: 3,330 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 71 72 73 74 75 76 77 78 79 80 81 | import Mathlib
import OAI.Analysis.HardSphere.Conditioning.ProductNormalize
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {I : Type*} [Fintype I] [DecidableEq I] {X : I→Type*} [∀i,MeasurableSpace (X i)]
lemma lintegral_ignore_coordinate (μ ν : ∀i,Measure (X i))
[∀i,IsProbabilityMeasure (μ i)] [∀i,IsProbabilityMeasure (ν i)]
(j : I) (he : ∀i,i≠j→μ i=ν i)
(F : (∀i,X i)→ℝ≥0∞) (hF : Measurable F)
(hignore : ∀x y,F (Function.update x j y)=F x) :
(∫⁻x,F x ∂Measure.pi μ)=∫⁻x,F x ∂Measure.pi ν := by
let x₀ : ∀i,X i := fun i ↦ Classical.choice (nonempty_of_isProbabilityMeasure (μ i))
rw [lintegral_eq_lmarginal_univ (μ:=μ) x₀,lintegral_eq_lmarginal_univ (μ:=ν) x₀,
lmarginal_erase' (μ:=μ) F hF (Finset.mem_univ j),lmarginal_erase' (μ:=ν) F hF (Finset.mem_univ j)]
simp_rw [hignore,lintegral_const,measure_univ,mul_one]
unfold lmarginal
congr 2
funext i
exact he i (Finset.mem_erase.mp i.property).1
lemma conditioned_lintegral_ignore (μ ν : ∀i,Measure (X i))
[∀i,IsProbabilityMeasure (μ i)] [∀i,IsProbabilityMeasure (ν i)]
(j : I) (he : ∀i,i≠j→μ i=ν i)
(A : Set (∀i,X i)) (hA : MeasurableSet A)
(hAi : ∀x y,Function.update x j y∈A↔x∈A)
(F : (∀i,X i)→ℝ≥0∞) (hF : Measurable F)
(hFi : ∀x y,F (Function.update x j y)=F x) :
(∫⁻x,F x ∂conditionedMeasure (Measure.pi μ) A)=
∫⁻x,F x ∂conditionedMeasure (Measure.pi ν) A := by
have hw (x : ∀i,X i) (y : X j) :
A.indicator (fun _ ↦ (1:ℝ≥0∞)) (Function.update x j y)=A.indicator (fun _ ↦ 1) x := by
by_cases hx : x∈A
· rw [indicator_of_mem ((hAi x y).mpr hx),indicator_of_mem hx]
· rw [indicator_of_notMem ((hAi x y).not.mpr hx),indicator_of_notMem hx]
have hm : (Measure.pi μ) A=(Measure.pi ν) A := by
have hh := lintegral_ignore_coordinate μ ν j he _ (measurable_const.indicator hA) hw
simpa only [lintegral_indicator_const hA,one_mul] using hh
rw [conditionedMeasure,conditionedMeasure,lintegral_smul_measure,lintegral_smul_measure,hm]
congr 1
rw [←lintegral_indicator hA,←lintegral_indicator hA]
apply lintegral_ignore_coordinate μ ν j he _ (hF.indicator hA)
intro x y
by_cases hx : x∈A
· rw [indicator_of_mem ((hAi x y).mpr hx),indicator_of_mem hx,hFi]
· rw [indicator_of_notMem ((hAi x y).not.mpr hx),indicator_of_notMem hx]
lemma conditioned_map_ignore {Y : Type*} [MeasurableSpace Y]
(μ ν : ∀i,Measure (X i))
[∀i,IsProbabilityMeasure (μ i)] [∀i,IsProbabilityMeasure (ν i)]
(j : I) (he : ∀i,i≠j→μ i=ν i)
(A : Set (∀i,X i)) (hA : MeasurableSet A)
(hAi : ∀x y,Function.update x j y∈A↔x∈A)
(f : (∀i,X i)→Y) (hf : Measurable f)
(hfi : ∀x y,f (Function.update x j y)=f x) :
(conditionedMeasure (Measure.pi μ) A).map f=
(conditionedMeasure (Measure.pi ν) A).map f := by
apply Measure.ext_of_lintegral
intro F hF
rw [lintegral_map hF hf,lintegral_map hF hf]
apply conditioned_lintegral_ignore μ ν j he A hA hAi _ (hF.comp hf)
intro x y
simp only [Function.comp_def,hfi]
end HardSphere.Comparison
end
end
end
end OAI
|