TITLE: If $f,g$ integrable then $f(x-y)g(y)$ integrable for almost every $x$
QUESTION [6 upvotes]: I am trying to prove that for two integrable functions $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$ the function $y \mapsto f(x-y)g(y)$ is integrable for almost every $x$. By using the holder inequality I reduced this to showing that if a function is integrable then also its square is integrable but after browsing a bit I found this so I guess this leads nowhere. Any hints are welcomed.

REPLY [8 votes]: Hint: consider $\int\int |f(x-y)g(y)| \, dy \, dx$, and use Tonelli's theorem to reverse the order of integration. If you can show this integral is finite, then $\int |f(x-y)g(y)| \, dy$ is finite for almost every $x$.