I am trying to prove the following statement.
if $f\in L^2$ and $g\in L^1$, $\mathcal{F}$ denotes fourier transformation,and$*$ denotes convolution, then $\mathcal{F}(f*g)=\mathcal{F}(f)\mathcal{F}(g)$
It should be extended from the case $f,g\in L^1$.
I followed the procedure starting from the lower part of page 72, but I couldn't understand the last estimation
$\int\int|f(x-y)-f_n(x-y)||g(y)|\,dx\,dy\\\le||f-f_n||_2||g||_1$
Any hints? Thanks!