Exercise 7, § 1

Let G be a locally compact group, $\beta$ a left Haar measure on G, A a subset of G, and B a relatively compact $\beta$-integrable subset of G such that $\beta(B) > 0$. Show that if the compact subsets of $A \cdot B$ have bounded measures for $\beta$, then $A$ is relatively compact. (Imitate the argument of Prop. 2.)