User:Caliburn/s/mt/Measure of Horizontal Section gives Measurable Function

Theorem

Let $\struct {X, \Sigma_X, \mu_X}$ and $\struct {Y, \Sigma_Y, \mu_Y}$ be $\sigma$-finite measure spaces.

Let $E \in \Sigma_X \otimes \Sigma_Y$.

Define the function $f : Y \to \overline \R$ by:

$\map f y = \map {\mu_X} {E^y}$

for each $y \in Y$.

Then $f$ is $\Sigma_Y$-measurable.