Let $(X, \mathcal{A} )$ and $(Y, \mathcal{B} )$ be measurable spaces. For any $E \in \mathcal{A} \times \mathcal{B} $, the sections $E_x$ and $E^y$ are measurable for any $x \in X$ and $y \in Y$.

Let $(X, \mathcal{A} )$ and $(Y, \mathcal{B} )$ be measurable spaces. Let $f: X \times Y \to F$, where $F$ is the extended real numnbers or the complex numbers, and $f$ is measurable (using the appropriate sigma algebra of the codomain). The sections $f_x: Y \to F$ and $f^y: X \to F$ are measurable for each $x \in X$ and $y \in Y$.