Let $(X, \Sigma_X)$, $(Y, \Sigma_Y)$ and $(Z, \Sigma_Z)$ be measurable spaces and consider a mapping $f : X \times Y \to Z$. The following sufficient condition for the measurability of the sections is well-known:
Suppose $f : X \times Y \to Z$ is $(\Sigma_X \otimes \Sigma_Y, \Sigma_Z)$-measurable. Then the sections $$f^x : (Y, \Sigma_Y) \to (Z, \Sigma_Z)$$ and $$f^y : (X, \Sigma_X) \to (Z, \Sigma_Z)$$ are $(\Sigma_Y, \Sigma_Z)$-measurable for each $x \in X$ respectively $(\Sigma_X, \Sigma_Z)$-measurable for each $y \in Y$.
The converse of the above result does not hold in general. However, I was curious whether there are conditions on the measurable spaces such that the converse implication does hold.
Any comment or reference is highly appreciated.