| import Mathlib |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| variable {W V : Type*} [MeasurableSpace W] [MeasurableSpace V] |
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| def fiberRestrict (κ : Kernel W V) [IsSFiniteKernel κ] (C : Set (W×V)) : Kernel W V := |
| κ.withDensity (fun w v ↦ C.indicator (fun _ ↦ (1 : ℝ≥0∞)) (w,v)) |
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| lemma fiberRestrict_apply (κ : Kernel W V) [IsSFiniteKernel κ] {C : Set (W×V)} |
| (hC : MeasurableSet C) (w : W) : |
| fiberRestrict κ C w=(κ w).restrict {v | (w,v)∈C} := by |
| have hm : Measurable (Function.uncurry (fun w : W ↦ fun v : V ↦ C.indicator (fun _ ↦ (1 : ℝ≥0∞)) (w,v))) := by |
| change Measurable (C.indicator (fun _ ↦ (1 : ℝ≥0∞))) |
| exact measurable_const.indicator hC |
| rw [fiberRestrict,Kernel.withDensity_apply _ hm] |
| have he : (fun v ↦ C.indicator (fun _ ↦ (1 : ℝ≥0∞)) (w,v))= |
| {v | (w,v)∈C}.indicator (fun _ ↦ (1 : ℝ≥0∞)) := rfl |
| rw [he] |
| have hs : MeasurableSet {v : V | (w,v)∈C} := hC.preimage (measurable_const.prodMk measurable_id) |
| exact withDensity_indicator_one hs |
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| def fiberNormalize (κ : Kernel W V) [IsSFiniteKernel κ] (C : Set (W×V)) : Kernel W V where |
| toFun w := (fiberRestrict κ C w univ)⁻¹ • fiberRestrict κ C w |
| measurable' := by |
| refine Measure.measurable_of_measurable_coe _ fun B hB ↦ ?_ |
| simp only [Measure.smul_apply,smul_eq_mul] |
| exact ((fiberRestrict κ C).measurable_coe MeasurableSet.univ).inv.mul |
| ((fiberRestrict κ C).measurable_coe hB) |
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| instance fiberNormalize_finite (κ : Kernel W V) [IsSFiniteKernel κ] (C : Set (W×V)) : |
| IsFiniteKernel (fiberNormalize κ C) := ⟨1,by simp,fun w ↦ by |
| change ((fiberRestrict κ C w univ)⁻¹ • fiberRestrict κ C w) univ≤1 |
| rw [Measure.smul_apply,smul_eq_mul] |
| exact ENNReal.inv_mul_le_one _⟩ |
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| lemma fiberNormalize_mass (κ : Kernel W V) [IsFiniteKernel κ] {C : Set (W×V)} |
| (hC : MeasurableSet C) {w : W} (hw : 0<κ w {v | (w,v)∈C}) : |
| fiberNormalize κ C w univ=1 := by |
| change ((fiberRestrict κ C w univ)⁻¹ • fiberRestrict κ C w) univ=1 |
| rw [fiberRestrict_apply κ hC,Measure.smul_apply,smul_eq_mul,Measure.restrict_apply_univ] |
| exact ENNReal.inv_mul_cancel hw.ne' (measure_ne_top _ _) |
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| lemma fiberNormalize_ac (κ : Kernel W V) [IsFiniteKernel κ] {C : Set (W×V)} |
| (hC : MeasurableSet C) (w : W) : fiberNormalize κ C w ≪ κ w := by |
| change (fiberRestrict κ C w univ)⁻¹ • fiberRestrict κ C w ≪ κ w |
| rw [fiberRestrict_apply κ hC] |
| exact Measure.smul_absolutelyContinuous.trans Measure.restrict_le_self.absolutelyContinuous |
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| lemma fiberNormalize_fst (μ : Measure W) [SFinite μ] (κ : Kernel W V) [IsFiniteKernel κ] |
| {C : Set (W×V)} (hC : MeasurableSet C) |
| (hpos : ∀ᵐ w ∂μ,0<κ w {v | (w,v)∈C}) : |
| (μ.compProd (fiberNormalize κ C)).map Prod.fst=μ := by |
| ext B hB |
| rw [Measure.map_apply measurable_fst hB,Measure.compProd_apply (hB.preimage measurable_fst)] |
| have he : (fun w ↦ fiberNormalize κ C w ((fun v ↦ (w,v))⁻¹' (Prod.fst⁻¹' B)))=ᵐ[μ] |
| B.indicator (fun _ ↦ (1 : ℝ≥0∞)) := by |
| filter_upwards [hpos] with w hw |
| by_cases hb : w∈B |
| · have hs : (fun v : V ↦ (w,v))⁻¹' (Prod.fst⁻¹' B)=univ := by ext v; simp [hb] |
| rw [hs,fiberNormalize_mass κ hC hw,Set.indicator_of_mem hb] |
| · have hs : (fun v : V ↦ (w,v))⁻¹' (Prod.fst⁻¹' B)=∅ := by ext v; simp [hb] |
| rw [hs,measure_empty,Set.indicator_of_notMem hb] |
| rw [lintegral_congr_ae he,lintegral_indicator_const hB,one_mul] |
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| lemma fiberNormalize_compProd_mass (μ : Measure W) [IsProbabilityMeasure μ] |
| (κ : Kernel W V) [IsFiniteKernel κ] {C : Set (W×V)} (hC : MeasurableSet C) |
| (hpos : ∀ᵐ w ∂μ,0<κ w {v | (w,v)∈C}) : |
| μ.compProd (fiberNormalize κ C) univ=1 := by |
| have h := congrArg (fun ν : Measure W ↦ ν univ) (fiberNormalize_fst μ κ hC hpos) |
| simpa only [Measure.map_apply measurable_fst MeasurableSet.univ,preimage_univ,measure_univ] using h |
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| end HardSphere.Comparison |
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| end |
| end |
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| end |
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| end OAI |
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