My friend and I were studying for a preliminary exam in probability and came across the following problem.
Suppose $X$ and $Y$ are independent random variables. Prove that if $X+Y\in L^1$ then $X$ and $Y$ are in $L^1$.
We had the following idea: the fact that $X+Y\in L^1$ implies that
$$\int_A Xd\mu = \infty \Leftrightarrow \int_A Yd\mu = \infty$$
for any measurable set $A$. If $B$ is any set for which $\int_B Xd\mu < \infty$ then
$$\mathbb{E}(Y\mathbb{1}_B)<\infty$$
but independence should give us that $\mathbb{E}(Y\mathbb{1}_B) = \mathbb{E}(Y)\mu(B)$ which should imply that $\mathbb{E}(Y)$ is finite. We're not really sure if we're heading in the right direction and are seeking a better approach.