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On page 67 of Jacod and Protter, Probability Essentials, it is stated that:

Theorem 10.2 Let $f$ be measurable: $(E \times F, {\cal E} \otimes {\cal F}) \to (\mathbf R, {\cal R})$. For each $x \in E$ (respt. $y \in F$), the "section" $y \to f(x,y)$ (resp. $x \to f(x,y)$) is an $\cal F$-measurable (resp. $\cal E$-measurable) function.

Note: The converse to Theorem 10.2 is false in general.

What is a counterexample to the converse of Theorem 10.2?

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