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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.

Waaal
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  • If each of the two expectations is finite, both are equal; otherwise, they are both infinite. – Amir Feb 15 '24 at 17:50
  • I understand that identically distributed random variables have the same expectation, but I don't see how that justifies using Fubini when the expectations are both infinite – Waaal Feb 15 '24 at 18:07
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    I think Fubini–Tonelli theorem is called Fubini here. You may see the section on Fubini–Tonelli theorem in the link https://en.m.wikipedia.org/wiki/Fubini%27s_theorem – Amir Feb 15 '24 at 18:14
  • The assumption of integrability of $Z_1$ (needed for the application of Fubini) is missing both from the original question and from the solution you are interested in. – John Dawkins Feb 18 '24 at 17:04

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