integral_kernel_prod_left — Mathlib

English

Given a kernel η on α×β and a function f on β×γ, the integral over the η-fibre in the γ-coordinate yields a kernel-valued function on α×β that is measurable, under mild assumptions.

Русский

При наличии ядра η на α×β и функции f на β×γ, интеграл по фибре в γ-координате образует измеримую функцию ядра на α×β.

LaTeX

$$$\\text{StronglyMeasurable } f \\implies \\text{StronglyMeasurable } x \\mapsto \\int_{\\gamma} f(x,y) \\, dη(x,y)$$$

Lean4

theorem integral_kernel_prod_left ⦃f : β → α → E⦄ (hf : StronglyMeasurable (uncurry f)) :
    StronglyMeasurable fun y => ∫ x, f x y ∂κ y :=
  (hf.comp_measurable measurable_swap).integral_kernel_prod_right'

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