When is an invariant mean the limit of a Følner net?

Let G be a locally compact amenable group, TLIM ( ) the topological left-invariant means on , and 0 limit points of Følner-nets. I show that = unless is σ -compact non-unimodular, in which case ≠ . This improves 1970 result Chou 2009 Hindman Strauss. consider analogous problem for non-topological means, give short construction net converging to invariance “weakly but not strongly,” simplifying proof 2001 Rosenblatt Willis. conclude with discussion some open problems.