Exercise 25, § 1

We denote by $\mu_G$ the Haar measure given on G. Let $\mu$ be a complex measure on G.

a) The image of the Fourier transform of $\mu$ is finite if and only if $\mu$ is a linear combination of idempotent measures, that is to say, of measures $\nu$ such that $\nu *\nu =\nu$.

b) The support of $\mathscr{F}_G(\mu)$ is finite if and only if $\mu$ is a finite linear combination of measures $\widehat{x}\mu_G$, where $\widehat{x}\in \widehat{G}$.

c) We denote by $A_\mu$ the set of measures of the form $\widehat{x} \mu$ for $\widehat{x}\in \widehat{G}$. The set of measures $\nu \in A_\mu$ such that $\nu (G)\not= 0$ is finite if and only if there exists a closed subgroup H of G, a finite set K, characters $\widehat{x}_k$ for $k\in K$ and complex numbers $t_k$ for $k\in K$ such that

where $i$ denotes the canonical inclusion of H in G.