I have a question about the joint measurability of a function defined as below:
Let $(S, \Sigma)$ be a measurable space, and let $X$ and $Y$ be topological spaces with Borel $\sigma-$algebras. Suppose that $X$ is countable, and for each $x\in X$, $f^x=f(\cdot, x):S\rightarrow Y$ is measurable. Can one say $f(\cdot, \cdot): S\times X\rightarrow$ is jointly measurable?