Measure of Vertical Section of Measurable Set 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 $f : X \to \overline \R$ by:


 * $\map f x = \map {\mu_Y} {E_x}$

for each $x \in X$.

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