 Solution

Interpret existence of the expectations as finite expected values, equivalently $\mathbb{E}|X|<\infty$ and $\mathbb{E}|Y|<\infty$. First justify the summations rather than assuming that a signed double series can be rearranged. Independence and Tonelli theorem giveConsequently $XY$ is an integrable random variable, and Fubini theorem permits summing the signed terms in either order:If extended infinite expectations are allowed instead, the phrase “exist” needs additional care: expressions such as $0\cdot \infty$ do not define an expectation. The finite-expectation convention avoids that ambiguity.