Exercise 16, § 1

Let $G$ be a compact abelian group, $E$ a dense subset of $G$ that is stable under the law of composition of $G$. Assume that, for every bounded numerical function $f$ on $E$, there is defined a number $M(f)$ having the following property: for any two bounded functions $f, g$ on $E$, for any elements $a_i, b_k$ of $E$ ($1 \leq i \leq m, 1 \leq k \leq n$), and for any real numbers $\alpha_i, \beta_k$ ($1 \leq i \leq m, 1 \leq k \leq n$) such that

for all $x \in E$, one then has the inequality

Assume in addition that $M(1) = 1$ (cf. TVS, IV, Appendix).

a) Let $\mu$ be the normalized Haar measure on $G$. Show that for every continuous numerical function $f$ on $G$, $\int f d\mu = M(f|E)$. (By considering a suitable partition of unity, approximate the constant function equal to $\int f d\mu$ on $E$ by means of functions of the form $x \mapsto \sum_i \alpha_i f(xa_i)$, where $a_i \in E$.)

b) Deduce from a) that if one sets $\nu(B) = M(\varphi_B)$ for every subset $B$ of $E$, then $\nu(U \cap E) \geq \mu(U)$ for every open subset $U$ of $G$, and $\nu(F \cap E) \leq \mu(F)$ for every closed subset $F$ of $G$. For every $\mu$-quadrable subset $P$ of $G$ (Ch. IV, §5, Exer. 17 d)), one has $\nu(P \cap E) = \mu(P)$.