Horizontal Section of Measurable Set is Measurable

Theorem
Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be measurable spaces.

Let $E \in \Sigma_X \otimes \Sigma_Y$ where $\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra of $\Sigma_X$ and $\Sigma_Y$.

Let $y \in Y$.

Then:


 * $E^y \in \Sigma_X$

where $E^y$ is the $y$-horizontal section of $E$.