Definition

Let $f:X\times Y\to \mathbb{C}$. Then for $x \in X,y\in Y$ we define $$\begin{align*} f_{x}:Y\to \mathbb{C},&\quad f_{x}(y)=f(x,y)\\ f_{y}:X\to \mathbb{C},&\quad f_{y}(x)=f(x,y) \end{align*}$$Then, suppose that $f$ is $\mathscr{P}$-measurable. Then $\forall x \in X,\forall y \in Y$: - $f_{x}$ is $\mathscr{N}$-measurable - $f^{y}$ is $\mathscr{M}$-measurable