Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 4 / b

For a finite generating set $S$, the Følner condition requires that for every $\epsilon >0$ there be a nonempty finite $F\subseteq G$ such thatwhere $△$ denotes symmetric difference. Choose a Følner sequence $(F_n)$ and define normalized counting functions on all subsets $A\subseteq G$ byBy compactness of the product $[0,1{]}^{\mathcal{P}(G)}$, some subnet converges pointwise to a function $m$. The identities ${\mu }_n(G)=1$ and finite additivity on disjoint subsets pass to the limit, so $m$ is a finitely additive probability measure.

For a fixed $g=s_1\cdots s_{\ell }$, the triangle inequality for symmetric differences givesConsequentlyso $m(gA)=m(A)$. The limit is left invariant and the Følner condition implies amenability. Thus every finitely generated group satisfying the Følner condition is amenable.