Let $f\in L^1(\mathbb R)$ and $g\in C_c^\infty(\mathbb R)$ such that $f\star g(x)=0$ a.e. Then show that $f=0$ a.e. Note, $g$ is a fixed function.
So from the given condition I can see $\hat{f}(y)\hat{g}(y)=0$ for every $y\in\mathbb R$ so $\hat{f}(y)=0$ or $\hat{g}(y)=0$ for every $y$.
I now thought of applying the uncertainty principle. The set $\{y:g(y)\neq 0\}$ is bounded, hence $\{y:\hat{g}(y)\neq 0\}$ must have infinite Lebesgue measure, in particular must be unbounded. So $\hat{f}(y)\neq0$ over a set with infinite measure. So what?
We can say much more about $\hat{g}$ than just that it is nonzero on an unbounded set. In particular, we can say that $\hat{g}$ is analytic (indeed, it is an entire function if we let its input range over $\mathbb{C}$). This means that if $\hat{g}$ is not identically $0$, its zeroes are isolated (and in particular form a set of measure $0$). So we must have $\hat{f}=0$ a.e., and so $f=0$ a.e.