If $X \perp Y$ and $X+Y \in L^1$ then $X,Y \in L^1$

If $X \perp Y$ and $X+Y \in L^1$ then $X,Y \in L^1$

Suppose $X,Y$ are independent random variables and that $X+Y \in L^1$. How
can I show that $X,Y \in L^1$.
The theorem obviously fails if independence is left off, but I don't see
how to use it. Intuitively, if $X$ failed to be in $L^1$, then it is big
somewhere, and the independence of $Y$ should says that $Y$ can't
selectively cancel out $X$ where $X$ is big.