When is an invariant mean the limit of a F{\o}lner net?

Let $G$ be a locally compact amenable group, $TLIM(G)$ the topological left-invariant means on $G$, and $TLI{M}_0(G)$ the limit points of Folner-nets. I show that $TLI{M}_0(G)=TLIM(G)$ unless $G$ is $\sigma$-compact non-unimodular, in which case $TLI{M}_0(G)\mathrm{\ne }TLIM(G)$. This improves a 1970 result of Chou and a 2009 result of Hindman and Strauss. I consider the analogous problem for the non-topological left-invariant means, and give a short construction of a net converging to invariance “weakly but not strongly,” simplifying the proof of a 2001 result of Rosenblatt and Willis.