Let $\widehat{\mu}$ be the Fourier transform of a Borel probability measure $\mu$ on $\mathbb R^d$. We look at the closed abelian subgroup $\Gamma (\mu)$ of $\mathbb R^d$, which consists of the periods of the function $\widehat{\mu}$. We prove the following dichotomy: i) The support of $\mu$ is non-degenerate if and only if $\Gamma (\mu)$ is a lattice. ii) The support of $\mu$ is degenerate if and only if $\Gamma (\mu)$ contains a linear subspace $\neq \{0\}$ of $\mathbb R^d$. A similar dichotomy is also discussed for the period group of the function $|\widehat{\mu}|$.