Let $(X, \mathcal{A}, \mu), (Y, \mathcal{B}, \nu)$ be $\sigma$-finite and complete measure spaces. $(X \times Y, \mathcal{A} \times \mathcal{B}, \mu \times \nu)$ is a (completion of) product measure space generated by $X$ and $Y$.
I wonder if the following is correct or not: Assume $f_n: X \times Y \to \mathbb{R}$ is $\mathcal{A} \times \mathcal{B}$-measurable, $f$ is a mapping from $X \times Y$ to $\mathbb{R}$, and "$a.e. y \in Y$, [$f_n(x, y) \to f(x, y), \, (a.e.x \in X)$]." More exactly,
$\exists N \in \mathcal{B}$ (null), $\, \forall y \in Y \setminus N,\, \exists M \in \mathcal{A}$ (null), $\, \forall x \in X \setminus M, \, f_n(x, y) \to f(x, y)$.
Then, $f$ is $\mathcal{A} \times \mathcal{B}$-measurable.
