Usually measure theory books include the following theorem (citing Proposition 5.1.3 in Cohn's measure theory book)
Let $(X, \mathcal A , \mu )$ and $(Y, \mathcal B, \nu )$ be $\sigma$-finite measure spaces. If $E$ belongs to the $\sigma$-algebra $\mathcal{A}\times\mathcal{B}$, then the function $x\mapsto\nu(E_x)$ is $\mathcal{A}$-measurable and the function $y\mapsto\mu(E^y)$ is $\mathcal{B}$-measurable.
where $E_x=\{y\in Y\mid (x,y)\in E\}$ and $E^y=\{x\in X\mid (x,y)\in E\}$ are the slices of $E$. I'm looking for an example of two spaces (with at least one of them necessarily not $\sigma$-finite) for which this theorem fails.
I believe one of the two needs to be not only $\sigma$-infinite but also s-infinite. If this is not the case I would be very interested in an example involving an s-finite but not $\sigma$-finite space.