Exercise 3, § 4

Let G be a locally compact group. a) Let $\mu$ be a nonzero bounded positive measure on G such that $\mu * \mu = \mu$. Show that the support S of $\mu$ is compact. (Let $f \in \mathscr{K}_+(G)$ with $f \neq 0$; choose a left Haar measure on G, with respect to which convolution products will be taken; we have $\mu * f \in \mathscr{K}(G)$; let $x \in S$ be such that $(\mu * f)(x) = \sup_{y \in S} (\mu * f)(y)$; show that $(\mu * f)(y) = (\mu * f)(x)$ for every $y \in S$.) b) Show that S is a compact subgroup of G and that $\mu$ is the normalized Haar measure of S. (Make use of a), Exer. 21 of GT, III, §4, and Exer. 5 of §3.)