Exercise 22, § 1

Let $G$ be a locally compact group, $\mu$ a left Haar measure on $G$, and $A$ an integrable subset of $G$ such that $\mu(A) > 0$. Show that the set $H(A)$ of $s \in G$ such that $\mu(A) = \mu(A \cap sA)$ is a compact group. (Observe that $H(A)$ is closed in $G$, with the help of Exer. 20. To see that $H(A)$ is compact, consider a compact subset $B$ of $A$ such that $\mu(B) > \mu(A)/2$ and prove that $H(A) \subset BB^{-1}$.)