Application of Fubini's Theorem 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 $L_1(\mathbb{R}^d)$.
 A: We want to think about $$\int_{\mathbb{R}^d} \left(\int_{\mathbb{R}^d} f(x-y)g(y)\;dy\right) e^{-i\xi\cdot x} \;dx.$$
The expression inside the parentheses is the convolution.  Suppose for now that we already know that $y \mapsto f(x-y)g(y)$ is in $L_1$.  The expression above is equal to
$$
\int_{\mathbb{R}^d} \left(\int_{\mathbb{R}^d} f(x-y)g(y)e^{-i\xi\cdot x}\;dy\right) \;dx
$$
(the factor $e^{-i\xi\cdot x}$ does not depend on $y$, so it is "constant").  Here we have an iterated integral.  Fubini's theorem says this is equal to the double integral
$$
\int_{\mathbb{R}^d\times\mathbb{R}^d} f(x-y)g(y)e^{-i\xi\cdot x} (dy\;dx)
$$
if the latter exists.  Since $x$ and $\xi$ are real, the factor $e^{-i\xi\cdot x}$ has absolute value $1$.  Hence we have
$$
\int_{\mathbb{R}^d\times\mathbb{R}^d} |f(x-y)g(y)e^{-i\xi\cdot x}| (dy\;dx) =
\int_{\mathbb{R}^d\times\mathbb{R}^d} |f(x-y)g(y)| (dy\;dx).
$$
All we need now is that that last integral is finite.  If you've seen a theorem saying $L_1$ is closed under convolution, you've got it.
Later note: If $L_1$ is closed under convolution, that says
$$
\int_{\mathbb{R}^d} |(f*g)(x)|\;dx < \infty,
$$
which is the same as saying that the iterated integral
$$
\int_{\mathbb{R}^d} \left(\int_{\mathbb{R}^d} |f(x-y)g(y)| \;dy\right)\;dx
$$
is finite.  That means where I wrote "...you've got it" above, you'd probably need to cite one more fact about integrals.
