Exercise 5, § 3

Let $G$ be a compact group, and $\mu$ a positive measure on $G$, with support $G$, such that $\mu * \mu = \mu$. Show that $\mu$ is the normalized Haar measure of $G$. (First show that $\| \mu \| = 1$. Then, if $\mu$ is not the Haar measure of $G$, there exists a function $f \in \mathscr{K}_+(G)$ such that

for all $t \in G$, and such that $\int f(s)\ d\mu(s) > \int f(st)\ d\mu(s)$ for some $t$. Show that one then has $(\mu * \mu)(f) > \mu(f)$.