Exercise 21, § 1

a) Let $G$ be a locally compact group, $\mu$ a left Haar measure on $G$. Show that if $S$ is a stable subset of $G$ such that $\mu_*(S) > 0$, then the interior of $S$ is nonempty (cf. Exer. 20). In particular, if a subgroup $H$ of $G$ has nonzero inner measure, then $H$ is an open subgroup of $G$.

b) If $G$ is compact, then every stable subset $S$ of $G$ such that $\mu_*(S) > 0$ is a compact open subgroup of $G$ (observe that the interior of $S$ is stable, and make use of Exer. 17 b)).

c) Assume that $G$ is compact and abelian, written additively, and that there exists in $G$ an element $a$ of infinite order. Show that there exists a stable subset $S$ of $G$ such that $a \in S, -a \notin S$ and $G = S \cup (-S)$ (use Zorn's lemma); deduce from b) that $S$ is not measurable and that $\mu_*(S) = 0$.