I am trying to prove the following seemingly obvious fact:
Let $\mu$ be a finite signed measure on $\mathbb R$. Suppose that $\hat\mu(u) = \int_\mathbb R e^{iux} d\mu(x) = 0$ for all $u$. Then $\mu(E) = 0$ for all measurable sets $E$.
However, I have not yet met with much success. First of all, is this actually true? And, if so, how can I prove it?