Exercise 10, § 4

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$. For any two $\beta$-integrable sets $A, B$ set $\rho(A, B) = \beta(A \cup B) - \beta(A \cap B)$. a) Let $A$ be a $\beta$-integrable set. Show that $x \mapsto \rho(xA, A)$ is a continuous function. b) Let $U$ be a neighborhood of $e$. Show that there exist a compact set $A$ and a number $\varepsilon > 0$ such that $\rho(xA, A) < \varepsilon$ implies $x \in U$. (Take for $A$ a neighborhood of $e$ such that $A \cdot A^{-1} \subset U .$) c) For a subset $C$ of $G$ to be relatively compact, it is necessary and sufficient that there exist a $\beta$-integrable set $A$ and a number $a$ ($0 < a < 2\beta(A)$) such that $x \in C$ implies $\rho(xA, A) \leq a$.