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a89d282 | 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 | import OAI.Probability.InvariantIsing.Pressure.RandomOrbitLaw
namespace OAI
/-! 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
end
end OAI
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