Exercise 8, § 1

Let $G$ be a locally compact group, $A$ a dense subset of $G$, $\alpha$ a left Haar measure on $G$, and $H$ an $\alpha$-measurable subset of $G$ having the following property: for every $s \in A$, $sH \cap (\mathbf{C}H)$ and $H \cap (\mathbf{C}sH)$ are locally $\alpha$-negligible. Show that either $H$ is locally $\alpha$-negligible or its complement is locally $\alpha$-negligible. (Show that $\varphi_H \cdot \alpha$ is left invariant.)