Let $\mu$ be *a finite complex valued measure* on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform
$$
\hat{\mu}(\omega)= \int\_{\mathbb{R}} e^{it\omega} d \mu(t)
$$
**Question:** Does $\hat{\mu}$ uniquely determine $\mu$? I am fairly sure that it does. However, I was not able to locate my standard references. Is there a good place where I can find proof of this fact?



Uniqueness of Fourier–Stieltjes transform for finite complex valued measures
I assume that $\mu$ is a regular complex Borel measure. Assume that $\widehat{\mu}=0$. Let $f \in \mathcal{S}(\mathbb{R})$ be a Schwartz class function. Then, writing $f=\widehat{g}$ for $g\in \mathcal{S}(\mathbb{R})$, we have from Fubini's theorem that
$$
\int\_{\mathbb{R}}f(t)d\mu(t) = \int\_{\mathbb{R}}\int\_{\mathbb{R}}g(s)e^{-its}dsd\mu(t) = \int\_{\mathbb{R}}g(s)\widehat{\mu}(-s)ds = 0.
$$
This implies $\mu$ is orthogonal to the Schwartz class, and by density, it's orthogonal to the space $C\_0(\mathbb{R})$ of continuous functions vanishing at infinity, which implies by Riesz's theorem that $\mu=0$.


