Exercise 52, § 1

Let H be a countable discrete group, not necessarily commutative. a) The group H is amenable (EVT, IV, p. 73, exercise 4) if and only if there exists an increasing sequence $(F_N)_N$ of finite subsets of H such that, for every $x\in H$, one has

lim$_{N\rightarrow+\infty}$ Card(F1 $_N$)Card((F$_N-xF_N$)$\cup (xF_N-F_N)$) $= 0$

(“Følner sequence”). If H is commutative, then it is amenable.

b) Let G be a compact commutative group. Fix a sequence $(F_N)_{N\geqslant 1}$ of finite subsets of the discrete group $\widehat{G}$ as in a) and denote by $\varphi_N$ the characteristic function of $F_N$. Put

$\psi_N=$ Card(F1 $_N$)$\varphi_N*\widetilde{\varphi}_N$.

The function $\psi_N$ has finite support, it satisfies $0\leqslant \psi_N\leqslant 1$, and

lim$_{N\rightarrow+\infty}\psi_N(\chi ) = 1$

for every $\chi \in \widehat{G}$.

c) Let $\mu_N$ be the measure on G of density the continuous mapping

The sequence of measures $(\mu_N)_{N\geqslant 1}$ converges towards $\varepsilon_e$ in the space $\mathscr{M}^1(G)$ endowed with the topology of compact convergence in $\mathscr{C}(G)$. (Apply Lemma 4 of INT, VIII, §2, n$^o7.$)

d) Let $f\in \mathscr{C}(G)$. One then has

$f=$ lim$_{N\rightarrow \infty}\sum_{\chi\in\widehat{G}}\psi_N(\chi )\mathscr{F}(f)(\chi )\chi$

in $\mathscr{C}(G)$.

e) Recover Fejér’s theorem (prop. 23 of II, p. 242) as a particular case of this statement.