Exercise 26, § 1

Let $G$ be a locally compact group, $\mu$ a left Haar measure on $G$, $k$ an integer $\geq 1$, $A$ an integrable set. Now prove that, for every $\varepsilon > 0$, there exists a neighbourhood $U$ of $e$ in $G$ having the following property: for every finite subset $S$ of $k$ elements in $U$, the set of $s \in G$ such that $sS \subset A$ has measure $\geq (1 - \varepsilon) \mu(A)$. (Reduce to the case where $A$ is compact and take $U$ such that

For every finite subset $H$ of $G$, put $p(H) = \mathrm{Card}(H)$ and $q(H) = 1$ if $H \neq \varnothing$, $q(H) = 0$ if $H = \varnothing$; by evaluating the integrals

prove that if, for $h \leq k$, one denotes by $M_h$ the set of the $s \in G$ for which $\mathrm{Card}(A \cap sS) = h$, one has

Conclude that $\sum_{h=1}^{k-1} h \mu(M_h) \leq k \varepsilon \cdot \mu(A)$.