Exercise 4, § 5

Let $G$ be a locally compact group that is not generated by any compact subset of $G$, and let $H$ be a discrete subgroup of $G$ such that $G/H$ is compact. Show that $H$ cannot be generated by a finite number of elements, but the normalized Haar measure $\alpha_0$ of $H$ belongs to the closure in $\Gamma^0$ of the set of $\alpha$ such that $\| \mu_\alpha \| = +\infty$ (consider the normalized Haar measures of the subgroups of $H$ that are generated by a finite number of elements).