Exercise 15, § 1

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$, so that $G$ operates continuously on $G/H$. Assume that $H$ satisfies the following condition: for every neighborhood $V$ of $e$ in $G$, there exists a neighborhood $U$ of $e$ such that $HU \subset VH$ (note that this condition is satisfied for every subgroup $H$ of $G$ if $e$ admits a fundamental system of neighborhoods invariant under every inner automorphism of $G$). Show that there then exists a nonzero positive measure on $G/H$ invariant under $G$. (Use the same method as in the proof of Th. 1 of No. 2, on observing, on the one hand, that as $U$ runs over the set of neighborhoods of $e$ in $G$, the images of the sets $HUH$ under the canonical mapping $\pi : G \to G/H$ form a fundamental system of neighborhoods of $\pi(H)$; on the other hand, for every neighborhood $V$ of $e$ in $G$ and every compact subset $K$ of $G$, there exist a neighborhood $U$ of $e$ in $G$ and a compact subset $L$ of $G$ such that every set of the form $sHUH$ that intersects $KH$ is of the form $s'HUH$ with $s' \in L$.)