| 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 | |