I am working with the same problem and have a follow up question to an old solution: https://math.stackexchange.com/a/1842770/1100158. Here they use Fubini's, but nowhere does it say that the random variables, $Z_1$ and $Z_2$, have finite expectation. Is there a reason why $E(Z_1I_{Z_1+Z_2\in A})$ is finite for any borel set $A\in\mathcal{B}(\mathbb{R})$? Clearly $A=\mathbb{R}$ with $|EZ_1|=\infty$ no longer works with this argument.
I ask because in the statement of my problem, we have defined expectations, so not necessarily finite.