For a finite generating set $S$, the <Følner condition> requires that for every $\varepsilon>0$ there be a nonempty finite $F\subseteq G$ such that
$$
\frac{|sF\mathbin\triangle F|}{|F|}<\varepsilon
\qquad(s\in S),
$$
where $\mathbin\triangle$ denotes <symmetric difference>. Choose a <Følner sequence> $(F_n)$ and define normalized counting functions on all subsets $A\subseteq G$ by
$$
\mu_n(A)=\frac{|A\cap F_n|}{|F_n|}.
$$
By 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 gives
$$
\frac{|gF_n\mathbin\triangle F_n|}{|F_n|}
\leq\sum_{j=1}^{\ell}
\frac{|s_jF_n\mathbin\triangle F_n|}{|F_n|}\longrightarrow0.
$$
Consequently
$$
|\mu_n(gA)-\mu_n(A)|
\leq\frac{|g^{-1}F_n\mathbin\triangle F_n|}{|F_n|}\longrightarrow0,
$$
so $m(gA)=m(A)$. The limit is left invariant and the <Følner condition implies amenability>. Thus \b[every finitely generated group satisfying the Følner condition is amenable].

Solution

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

{c}

Normalized counting measures on a <Følner sequence> have a convergent subnet in the compact product $[0,1]^{\mathcal P(G)}$. The limit is a left-invariant finitely additive probability measure, so every group satisfying the <Følner condition> is an <amenable group>.

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 Incoming links: Følner condition implies amenability