Uniqueness of Fourier transform in $L^1$

$$ \int_{\mathbb R} e^{-a|x|^2/2 + i x x_0} \hat f(x) \, dx = \int_{\mathbb R} f(y) \int_{\mathbb R} \frac1{2\pi} e^{-a|x|^2/2} e^{-ix(y-x_0)} \, dx \, dy = \int_{\mathbb R} \frac1{\sqrt{2 \pi a}} e^{-|y-x_0|^2/2a} f(y) \, dy .$$ The first equality is by substitution and Fubini. The second uses standard tables of Fourier transforms. The last quantity converges to $f(x_0)$ in $L_1$ as $a\to 0^+$ (see first that it converges if $f$ is a continuous and compactly supported, then use the usual tricks of approximating $f$ by a such a function). I might be off by a factor of $2\pi$.

EDIT: This only works for $f,g\in L^1\cap L^2$

HINT: Use the inverse fourier transform: $$f(x)=\frac{1}{2\pi}\int_\mathbb{R}\hat{f}(y)e^{ixy}dy$$

and let $\hat{h}=\hat{f}-\hat{g}\equiv0$, then conclude.