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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 | import Mathlib
import OAI.Analysis.HardSphere.Comparison.FiberNormalize
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {W V : Type*} [MeasurableSpace W] [MeasurableSpace V]
lemma fiberRestrict_le (κ : Kernel W V) [IsFiniteKernel κ] {C : Set (W×V)} (hC : MeasurableSet C) (w : W) :
fiberRestrict κ C w≤κ w := by rw [fiberRestrict_apply κ hC]; exact Measure.restrict_le_self
lemma exposed_mass_positive (μ : Measure W) (κ : Kernel W V) [IsFiniteKernel κ]
{C : Set (W×V)} (hC : MeasurableSet C) :
∀ᵐ w ∂μ.withDensity (fun w ↦ fiberRestrict κ C w univ),0<κ w {v | (w,v)∈C} := by
rw [ae_withDensity_iff ((fiberRestrict κ C).measurable_coe MeasurableSet.univ)]
exact ae_of_all _ fun w hw ↦ by
rw [fiberRestrict_apply κ hC,Measure.restrict_apply_univ] at hw
exact pos_iff_ne_zero.mpr hw
lemma fiber_rescale (κ : Kernel W V) [IsFiniteKernel κ] {C : Set (W×V)}
(hC : MeasurableSet C) (w : W) (B : Set V) :
(fiberRestrict κ C w univ) * (fiberNormalize κ C w B)=fiberRestrict κ C w B := by
change (fiberRestrict κ C w univ) * (((fiberRestrict κ C w univ)⁻¹ • fiberRestrict κ C w) B)=_
rw [Measure.smul_apply,smul_eq_mul,←mul_assoc]
by_cases hz : fiberRestrict κ C w univ=0
· have h0 : fiberRestrict κ C w=0 := Measure.measure_univ_eq_zero.mp hz
simp [h0]
· have hfin : fiberRestrict κ C w univ≠⊤ := ne_of_lt ((fiberRestrict_le κ hC w univ).trans_lt (measure_lt_top _ _))
rw [ENNReal.mul_inv_cancel hz hfin,one_mul]
lemma fiber_disintegrate (μ : Measure W) [SFinite μ] (κ : Kernel W V) [IsFiniteKernel κ]
{C : Set (W×V)} (hC : MeasurableSet C)
[SFinite (μ.withDensity (fun w ↦ fiberRestrict κ C w univ))] :
(μ.withDensity (fun w ↦ fiberRestrict κ C w univ)).compProd (fiberNormalize κ C)=
(μ.compProd κ).restrict C := by
ext B hB
rw [Measure.compProd_apply hB,lintegral_withDensity_eq_lintegral_mul₀]
· simp only [Pi.mul_apply]
simp_rw [fiber_rescale κ hC]
rw [Measure.restrict_apply hB,Measure.compProd_apply (hB.inter hC)]
apply lintegral_congr
intro w
have hBw : MeasurableSet (Prod.mk w ⁻¹' B) := hB.preimage (measurable_const.prodMk measurable_id)
rw [fiberRestrict_apply κ hC,Measure.restrict_apply hBw]
rfl
· exact ((fiberRestrict κ C).measurable_coe MeasurableSet.univ).aemeasurable
· exact ((fiberNormalize κ C).measurable_kernel_prodMk_left hB).aemeasurable
end HardSphere.Comparison
end
end
end
end OAI
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