  Let $G$ be a locally compact amenable group, $TLIM(G)$ the topological
left-invariant means on $G$, and $TLIM_0(G)$ the limit points of Folner-nets. I
show that $TLIM_0(G) = TLIM(G)$ unless $G$ is $\sigma$-compact non-unimodular,
in which case $TLIM_0(G) \neq TLIM(G)$. 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.
