Sum of Independent RVs in $L^1$

Question

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.

Answer 1

By independence $$ {\rm E}[|X+Y|]=\int_{\mathbb{R}^2}|x+y|P_{(X,Y)}(\mathrm dx,\mathrm dy)=\int_{\mathbb{R}^2}|x+y|P_X\otimes P_Y(\mathrm dx,\mathrm dy) $$ and hence by Tonelli $$ {\rm E}[|X+Y|]=\int_{\mathbb{R}} {\rm E}[|X+y|]P_Y(\mathrm dy). $$ If ${\rm E}[|X+Y|]<\infty$ then ${\rm E}[|X+y|]<\infty$ for $P_Y$-almost-all $y$. In particular, $X+y\in L^1(P)$ for some $y\in\mathbb{R}$. Thus also $X\in L^1(P)$.