Exercise 7, § 2

a) Let $E$ be a locally compact space, $\Gamma$ a locally compact group operating on the left continuously in $E$. Assume that for every $x \in E$, the mapping $s \mapsto s \cdot x$ of $\Gamma$ into $E$ is proper, and that there exists a nonzero bounded positive measure $\mu$ on $E$ invariant under $\Gamma$. Then $\Gamma$ is compact. (Let $(s_n)$ be a sequence of points of $\Gamma$. Let $K$ be a compact subset of $E$ that is not $\mu$-negligible. Show, using Exer. 15 of Ch. IV, §4, that there exists an $x_0 \in E$ such that $s_n x_0 \in K$ for infinitely many values of $n$. From this, deduce that $(s_n)$ has a cluster point in $\Gamma$. Then apply Exer. 6 of GT, II, §4.)

b) Let $G$ be a locally compact group, $H$ and $K$ two closed subgroups of $G$. Assume that $G$ and $H$ are unimodular and that $G/H$ has finite measure for the measure invariant under $G$. Show that for $H$ and $K$ to satisfy the equivalent conditions of Exer. 11 c) of GT, III, §4, it is necessary and sufficient that $K$ be compact.