Integrability of the function $f_1(y)f_2(x-y)$ for almost all $x$. (convolution)

Question

What I'm trying to prove is that for $f_1,f_2 \in \mathcal{L}_1$ the function $y \mapsto f_1(y)f_2(x-y)$ is integrable for almost all $x$, or:

$$ F(x) = \int f_1(y)f_2(x-y)\,dy < \infty \text{ almost all }x.$$

I've already shown that the convolution is integrable, but thanks to Fubini thats easy. Here there are less tools to use. I've also looked at Hölder's inequality, but nothing seems to quite work.

Any help/tips would be much appreciated!

Answer 1

Hint: look up Young's Inequality. I posted a proof here.