Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

Question

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$

Suppose $f$ is a continuous function with a compact support.

We note, by trivial, inequality, that , $\|\hat{f}\|_{L^{\infty}}\leq \|f\|_{L^{1}(\mathbb R)}.$

My Question is: Can we expect to identify, the Fourier transform of $f$, namely, $\hat{f}\in L^{\infty}(\mathbb R) $ with measure in $M(\mathbb R)$ ? If yes, how ?

Thanks,