Dense set in $L^2$ Let $ \Omega\subset \mathbb{R}^n$ with $m(\Omega^c)=0 $. Then how can we show that $ \mathcal{F}(C_{0}^{\infty}(\Omega))$ (here $ \mathcal{F}$ denotes the fourier transform) is dense in $L^2$(or $L^p$)?
Besides,  I'm also interested to know if the condition that $m(\Omega^c)=0$ can be weakened to some more general set.
Thanks for your help.

To deal with the case $\Omega=\Bbb R^n$, take $f\in L^2$. Then by Plancherel's theorem, theorem 12 in these lecture notes, we can find $g\in L^2$ such that $f=\mathcal F g$. Now approximate $g$ by smooth functions with compact support and use isometry. 
