ae_of_compProd — Mathlib

English

Under S-finite hypotheses, the AEStronglyMeasurable of f on μ.compProd κ implies equality of integrals with sliced measurability.

Русский

Под условия S-финитности AEStronglyMeasurable f на μ.compProd κ эквивалентно равенству интегралов по срезам.

LaTeX

$$$$ \\mathrm{AEStronglyMeasurable}\,f\, (μ.compProd κ) \\Rightarrow (\\int f dμ.compProd κ = \\int a \\int b f(a,b) dκ(a) dμ(a)) $$$$

Lean4

theorem _root_.MeasureTheory.AEStronglyMeasurable.ae_of_compProd [SFinite μ] [IsSFiniteKernel κ] {E : Type*}
    [NormedAddCommGroup E] {f : α → β → E} (hf : AEStronglyMeasurable f.uncurry (μ ⊗ₘ κ)) :
    ∀ᵐ x ∂μ, AEStronglyMeasurable (f x) (κ x) := by simpa using hf.compProd_mk_left

Похожие утверждения