Left Følner nets for locally compact groups

For every $g\in G$, [A1] gives $\mu (gF)=\mu (F)$, so $\mu (gF\mathrm{△}F)\le \mu (gF)+\mu (F)=2\mu (F)$. Thus each ratio in ${\mathrm{\Delta }}_Q(F)$ is in $[0,2]$ and the displayed supremum is a finite real; including $0$ also defines it when $Q$ is empty. If the condition has been checked for compact sets containing $e$, then for arbitrary compact $Q$ apply it to $Q\cup \{e\}$, which is compact as a finite union of compact sets; the resulting estimate restricts to $Q$. The reverse implication is immediate.

If $(F_i{)}_{i\in I}$ is a left Følner net, then for any compact $Q$ and $\epsilon >0$ its defining uniform-convergence condition supplies an index $i_0$ with ${\mathrm{\Delta }}_Q(F_i)\le \epsilon$ for every $i⪰i_0$. In particular $F_{i_0}$ is Borel, has finite positive measure, and satisfies the single-set Følner estimate.

Conversely, assume the single-set condition. Let $I$ be the set of all triples $(Q,\epsilon ,F)$ with $Q$ compact, $\epsilon >0$, $F$ Borel, $0<\mu (F)<\mathrm{\infty }$, and ${\mathrm{\Delta }}_Q(F)\le \epsilon$. Order these triples by $(Q,\epsilon ,F)⪯(Q^{\mathrm{\prime }},{\epsilon }^{\mathrm{\prime }},F^{\mathrm{\prime }})$ exactly when $Q\subseteq Q^{\mathrm{\prime }}$ and ${\epsilon }^{\mathrm{\prime }}\le \epsilon$. This is a directed preorder: for two indices apply the condition to the compact union of their test sets and the positive minimum of their tolerances, obtaining a witness that gives a common upper bound. By [F1], the third-coordinate map $i\mapsto F_i$ is a net. Given any compact $Q$ and $\epsilon >0$, the condition supplies an index $i_0=(Q,\epsilon ,F_0)$; every $i⪰i_0$ then satisfies ${\mathrm{\Delta }}_Q(F_i)\le {\mathrm{\Delta }}_{Q_i}(F_i)\le {\epsilon }_i\le \epsilon$. This proves uniform convergence on compact sets. The witness-indexed set contains every possible witness, so this construction uses no global choice function.