| import OAI.Probability.InvariantIsing.Pressure.RandomOrbitLaw |
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|
| namespace OAI |
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|
| /-! Integration against the joint conditional orbit law does not require a |
| (global) integrability hypothesis for the random pressure. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| namespace InvariantIsing |
|
|
| lemma ConditionalFieldOrbitLaw.lintegral {Ω : Type*} [MeasurableSpace Ω] {N : ℕ} |
| {P : Measure Ω} [IsProbabilityMeasure P] {d : Ω → FieldSpectralData N} {Y : Ω → ℝ} |
| {H : Measure (Orthogonal N)} [IsProbabilityMeasure H] |
| (h : ConditionalFieldOrbitLaw P d Y H) (hN : 0 < N) |
| (hd : Measurable d) (hY : Measurable Y) |
| (φ : FieldSpectralData N × ℝ → ℝ≥0∞) (hφ : Measurable φ) : |
| (∫⁻ ω, φ (d ω,Y ω) ∂P)= |
| ∫⁻ ω, ∫⁻ U, φ (d ω,dataPhysicalPressure (d ω) U) ∂H ∂P := by |
| have hm : Measurable (fun z : FieldSpectralData N × Orthogonal N => |
| (z.1,dataPhysicalPressure z.1 z.2)) := measurable_fst.prodMk (measurable_dataPhysicalPressure hN) |
| have hF : Measurable (fun z : FieldSpectralData N × Orthogonal N => |
| φ (z.1,dataPhysicalPressure z.1 z.2)) := hφ.comp hm |
| rw [← lintegral_map hφ (hd.prodMk hY),h,lintegral_map hφ hm, |
| lintegral_prod _ hF.aemeasurable] |
| exact lintegral_map hF.lintegral_prod_right' hd |
|
|
| end InvariantIsing |
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| end |
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| end OAI |
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|
