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.

Since $X$ and $Y$ are independent, $E[|X+Y|]=E[u(X)]$ where $u(x)=E[|x+Y|]$. If $X+Y$ is integrable, then $u(X)$ is integrable, in particular, $u(X)$ is almost surely finite, thus there exists at least one value $x$ such that $u(x)$ is finite. Then $x+Y$ is integrable, hence $Y$ is integrable. Finally, $X=(X+Y)+(-Y)$ is integrable as a sum of integrable random variables.
