Følner sequence

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