Exercise 26, § 1

Let G be a locally compact group, $\mu$ a left Haar measure on G, $k$ an integer $\geq 1$, and A an integrable set. Show that for every $\varepsilon > 0$, there exists a neighborhood 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 that A is compact and take U to be such that

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

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

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