Følner condition implies amenability

Remarks

Each $m_n$ is positive and satisfies $m_n(\mathbf{1}) = 1$. For $g \in G$ and $(g \cdot f)(x) = f(g^{-1} x)$, one has $m_n(g \cdot f) = \frac{1}{\abs{F_n}} \sum_{y \in g^{-1} F_n} f(y)$, so $$\abs{m_n(g \cdot f) - m_n(f)} \le \norm{f}_\infty \frac{\abs{g^{-1} F_n \triangle F_n}}{\abs{F_n}} \to 0.$$ The means form a weak-$*$ compact set by the Banach–Alaoglu theorem, so $(m_n)$ has a cluster point $m$, and the estimate shows that $m$ is invariant.

Example: the intervals $\set{0, \ldots, n-1}$ in $\Z$ give invariant means on $\Z$; restricted to functions supported on $\N$, they are Banach limits obtained from Cesàro averages.

The converse also holds: Følner's theorem says that every countable amenable group has a Følner sequence. The same argument works for a locally compact group with a Følner sequence of compact sets, using normalized Haar measure on $F_n$.

Going deeper

For a student

This is the easy half of Følner's characterization of amenability, and it is the usual way to prove that a given group is amenable. For example, groups of polynomial growth, and in particular finitely generated nilpotent groups, have Følner sequences made of balls. The limit step is not constructive: it uses compactness, and the resulting mean is not unique in general.