Følner condition

A finitely generated group $G$ with finite generating set $S$ satisfies the Følner condition when, for every $\epsilon >0$, there is a nonempty finite subset $F\subseteq G$ such thatThe condition is independent of the finite generating set.

A Følner sequence is a sequence of nonempty finite subsets $F_n\subseteq G$ such that $|g{F}_n△F_n|/|F_n|\to 0$ for every $g\in G$.

For a finitely supported nonnegative function $\mu$ of sum one, let $F_t=\{x:\mu (x)>t\}$. Then ${\int }_0^{\infty }|F_t|dt=1$ and ${\int }_0^{\infty }|F_t△g{F}_t|dt=\Vert \mu -\lambda (g)\mu {\Vert }_1$. If the sum of these latter integrals over a finite set of translations is less than $\epsilon$, some nonempty level set has the sum of its relative boundaries less than $\epsilon$. This converts the Reiter condition into finite Følner sets without assuming that the group is finitely generated.

The intervals $F_n=\{-n,\dots ,n\}$ form a Følner sequence for the additive group $\mathbb{Z}$, because

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.