This question is inspired by this other question. Recall that if $\Omega \subset \Bbb{R}^N$ is an open set, then $f : \overline \Omega \times \Bbb{R} \longrightarrow \Bbb{R}$ is Catahéodory if
(a) for all $s \in \Bbb{R}$ the map $x \mapsto f(x, s)$ is Lebesgue measurable;
(b) for almost every $x \in \overline \Omega$ the map $s \mapsto f(x, s)$ is continuous.
Then, how do we show that $f$ is measurable on the product $\sigma$-algebra $\mathcal{L} \otimes \mathcal{B}$, the product of the Lebesgue $\sigma$-algebra on $\Bbb{R}^N$ and the Borel $\sigma$-algebra on $\Bbb{R}$?
As can be seen in the question linked, I tried addapting an exercise of Folland's Real Analysis, but to no end. The main difficulty is that $s \mapsto f(x, s)$ is contnuous only almost everywhere in $\overline \Omega$.
Thanks in advance and kind regards.