Uniqueness of Fourier transform of a measure

Question

Let $C_0(\mathbb{R}^d)$ be the space of real-valued continuous functions on $\mathbb{R}^d$ that vanish at infinity. By Riesz–Markov–Kakutani representation theorem, the dual of $C_0(\mathbb{R}^d)$ is $\mathcal{M}(\mathbb{R}^d)$ composed of all the real-valued regular Borel measures on $\mathbb{R}^d$. For any $\mu\in\mathcal{M}(\mathbb{R}^d)$, the Fourier transform of $\mu$ is defined by $$ \hat\mu(\xi)=\int_{\mathbb{R}^d}e^{-ix^\top\xi}d\mu(x),\quad\xi\in\mathbb{R}^d. $$ My question is, if $\hat\mu(\xi)=0$ for each $\xi\in\mathbb{R}^d$, could we draw the conclusion that $\mu$ is zero measure? I find some literature saying that it is true when $\mu$ is a finite measure, namely $\mu(\mathbb{R}^d)<\infty$. So are all the measures in $\mathcal{M}(\mathbb{R}^d)$ finite?