| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter ProbabilityTheory | |
| open scoped ENNReal NNReal Topology BigOperators | |
| namespace HardSphere.Measure | |
| lemma tprod_ac {ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)] | |
| (μ ν : ∀i,Measure (X i)) [∀i,SigmaFinite (μ i)] [∀i,SigmaFinite (ν i)] | |
| (h : ∀i,μ i ≪ ν i) (l : List ι) : Measure.tprod l μ ≪ Measure.tprod l ν := by | |
| induction l with | |
| | nil => rfl | |
| | cons i l ih => | |
| rw [Measure.tprod_cons, Measure.tprod_cons] | |
| exact (h i).prod ih | |
| lemma pi_ac {ι : Type*} [Fintype ι] {X : ι → Type*} [∀ i, MeasurableSpace (X i)] | |
| (μ ν : ∀i,Measure (X i)) [∀i,SigmaFinite (μ i)] [∀i,SigmaFinite (ν i)] | |
| (h : ∀i,μ i ≪ ν i) : Measure.pi μ ≪ Measure.pi ν := by | |
| classical | |
| let : Encodable ι := Fintype.toEncodable ι | |
| rw [←Measure.pi'_eq_pi μ,←Measure.pi'_eq_pi ν,Measure.pi',Measure.pi'] | |
| simpa only [MeasurableEquiv.piMeasurableEquivTProd_symm_apply] using | |
| (MeasurableEquiv.piMeasurableEquivTProd (Encodable.sortedUniv_nodup ι) (Encodable.mem_sortedUniv)).symm.measurableEmbedding.absolutelyContinuous_map | |
| (tprod_ac μ ν h (Encodable.sortedUniv ι)) | |
| lemma ac_map {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] {μ ν : Measure X} | |
| {f : X→Y} (hf : Measurable f) (h : μ ≪ ν) : μ.map f ≪ ν.map f := by | |
| refine Measure.AbsolutelyContinuous.mk fun S hS hnull ↦ ?_ | |
| rw [Measure.map_apply hf hS] at hnull ⊢ | |
| exact h hnull | |
| end HardSphere.Measure | |
| end | |
| end | |
| end | |
| end OAI | |