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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 | 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
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