Exercise 9, § 4

Let G be a locally compact group, $\beta$ a left Haar measure on G, A and B two $\beta$-integrable sets such that $\beta(A) \leq \beta(B)$ and $\beta^*(A^{-1}) < +\infty$. Show that there exist 1) disjoint sets N, K_1, K_2, ... covering A, with N $\beta$-negligible and the K_n compact;

2) disjoint sets $N' , K'_1 , K'_2 , \ldots$ covering B, with the $K'_n$ compact; 3) elements $s_n \in G$ such that $K'_n = s_n K_n$. (Using the fact that $\beta(xA \cap B)$ depends continuously on $x$ and that

show that if $\beta(A) \neq 0$ then there exists an $x \in G$ such that $\beta(xA \cap B) \neq 0 .$)