There is a pretty classic question that is:
Let $X=Y=[0,1]$, $\mathcal{M}=\mathscr{B}_{[0,1]}$, $\mathcal{N}=2^{[0,1]}$, let $\mu$ be Lebesgue measure on $\mathcal{M}$ and let $\nu$ be counting measure on $\mathcal{N}$. Let $\Delta$ denote the diagonal $\Delta=\{(x,x)|x\in[0,1]\}\subset[0,1]\times[0,1]$. Prove that $\Delta\in\mathcal{M}\otimes\mathcal{N}$ and $$\int_X(\int_Y1_E(x,y)d\nu(y))d\mu(x),\int_Y(\int_X1_E(x,y)d\mu(x))d\nu(y)$$ are unequal. Show that, for each $P\in\mathcal{M}\otimes\mathcal{N}$, the functions $f(x)=\nu(P_x)$ and $g(y)=\mu(P^y)$ are measurable and $\tau(P)=\int_X\nu(P_x)d\mu$, $\rho(P)=\int_Y\mu(P^y)d\nu$ are both measures on $\mathcal{M}\otimes\mathcal{N}$.
The actual thing I am having a problem with is showing that $f(x)$ is measurable. My instinct is to show that the set for which this is true is a sigma algebra that contains sets of the form $E\times F$ where $E\in\mathcal{M}$ and $F\in\mathcal{N}$, but its not quite clear to me how to actually accomplish this.