I am trying to show that for $f,g\in L_1(\mathbb{R}^d)$, $f*g\in L_1(\mathbb{R}^d)$.
Somewhere along the way I need to switch the order of integration in the following integral (I know this for sure because it is literally a step out of my professor's notes).
$\int_{\mathbb{R}^{d}}f(x-y)g(y)e^{-i\xi\cdot x}\;dy\;dx.$
In the notes it says "By Fubini's Theorem". But I can't verify the hypothesis of the theorem which says the integrals may be switched if the following is true:
$f(\cdot-y)g(y)e^{-i\xi\cdot \cdot}$ and $f(x-\cdot)g(\cdot)e^{-i\xi\cdot x}$ are both in $L_1(\mathbb{R}^d)$.