| 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 A B : Type*} [MeasurableSpace W] [MeasurableSpace A] [MeasurableSpace B] |
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| lemma integrated_primary_density (ξ : Measure W) [SFinite ξ] |
| (κ : Kernel W A) (η : Kernel W B) [IsSFiniteKernel κ] [IsSFiniteKernel η] |
| (f : A→B) (hf : Measurable f) (L : W×B→ℝ≥0∞) (hL : Measurable L) |
| (h : ∀ᵐw ∂ξ,(κ w).map f=(η w).withDensity (fun b ↦ L (w,b))) : |
| (ξ.compProd κ).map (fun p ↦ (p.1,f p.2))=(ξ.compProd η).withDensity L := by |
| apply Measure.ext_of_lintegral |
| intro F hF |
| have hm : Measurable (fun p : W×A ↦ (p.1,f p.2)) := measurable_fst.prodMk (hf.comp measurable_snd) |
| rw [lintegral_map hF hm,Measure.lintegral_compProd (show Measurable (fun p : W×A ↦ F (p.1,f p.2)) from hF.comp hm), |
| lintegral_withDensity_eq_lintegral_mul _ hL hF, |
| Measure.lintegral_compProd (hL.mul hF)] |
| apply lintegral_congr_ae |
| filter_upwards [h] with w hw |
| rw [←lintegral_map (show Measurable (fun b : B ↦ F (w,b)) from hF.comp measurable_prodMk_left) hf,hw, |
| lintegral_withDensity_eq_lintegral_mul _ |
| (show Measurable (fun b : B ↦ L (w,b)) from hL.comp measurable_prodMk_left) |
| (show Measurable (fun b : B ↦ F (w,b)) from hF.comp measurable_prodMk_left)] |
| rfl |
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| lemma map_density_comp (μ : Measure A) (f : A→B) (hf : Measurable f) |
| (L : B→ℝ≥0∞) (hL : Measurable L) : |
| (μ.withDensity (L∘f)).map f=(μ.map f).withDensity L := by |
| apply Measure.ext_of_lintegral |
| intro F hF |
| rw [lintegral_map hF hf] |
| change (∫⁻a,(F∘f) a ∂μ.withDensity (L∘f))=_ |
| rw [lintegral_withDensity_eq_lintegral_mul _ (hL.comp hf) (hF.comp hf), |
| lintegral_withDensity_eq_lintegral_mul _ hL hF,lintegral_map (hL.mul hF) hf] |
| rfl |
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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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