Exercise 11, § 2

Let G be a locally compact group, H a closed subgroup of G, $\lambda$ a left Haar measure on G; assume that there exists on G/H a measure $\mu$ invariant under G such that $\lambda = \mu^\#$ and $\mu(G/H) < +\infty$. Let $\nu$ be a nonzero positive measure on G/H such that $\nu(G/H) < +\infty$. Finally, let $h$ be a function on $G$ having the properties of No. 4, Prop. 8.

a) Let $A$ be a Borel subset of $G/H$. Show that

(make use of Prop. 9 a) of No. 4).

b) Let $A_i$ ($1 \leq i \leq n$) be Borel subsets of $G/H$ and, for every $i$, let $b_i = \sup_{s \in G} \nu(s^{-1}A_i)$. Show that there exist two distinct indices $i, j$ such that

(argue as in §1, Exer. 28 a)).