The problem statement:
Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes \nu)$ and $$\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu.$$
I did the usual routine: let first $f,g$ be indicator functions, then simple functions, then non-negative measurable, then integrable.
My question is: where does one (implicitly) use the $\sigma$-finiteness of $\mu, \nu$? I went throw the argument and couldn't spot the place. Since this was a stand-alone exercise, I though it's rather unlikely for the setup to be redundant.
$\mu \otimes \nu$ is only well-defined when $\mu$ and $\nu$ are $\sigma$-finite. – Stefan Hansen