Let $(X,\mu_x)$ and $(Y,\mu_y)$ be two measure spaces endowed with $\sigma$-additive compete measures $\mu_x$ and $\mu_y$, respectively. Let $\mu:=\mu_x\otimes\mu_y$ be the Lebesgue extension of measure $\mu_x\times\mu_y$ defined by $(\mu_x\times\mu_y)(A\times B)=\mu_x(A)\mu_y(B)$ for any two measurable sets $A\subset X$, $B\subset Y$.
I wonder what we can say about the the $\mu$-measurability of a function $f:X\times Y\to\mathbb{K}$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, if we know the $\mu_x$-measurability of $f(-,y):X\to\mathbb{K}$ for all $y\in Y$ and the $\mu_y$-measurability of $f(x,-):Y\to\mathbb{K}$ for all $x\in X$, and vice versa. Does any implication between the two exist?
The issue has arisen in my mind because I found the statement in problem 6 here that if $\int_X (\int_{A_x}|f(x,y)| d\mu_y)d\mu_x$, where $A_y=\{y\in Y:(x,y)\in A\}$, exists then $\int_A fd\mu$ does. I think it is implicitly intended that $f:A\to\mathbb{K}$ is measurable, but I think it would be interesting to explore the possibility of reciprocal implications. Thank you very much for any answer!