AbstractLet with . If G is a nondiscrete locally compact group which is amenable as a discrete group and m ∈ LIM(CB(G)), then we can embed into the set of all extensions of m to left invariant means on L∞(G) which are mutually singular to every element of TLIM(L∞(G)), where LIM(S) and TLIM(S) are the sets of left invariant means and topologically left invariant means on S with S = CB(G) or L∞(G). It follows that the cardinalities of LIM(L∞(G)) ̴ TLIM(L∞(G)) and LIM(L∞(G)) are equal. Note that which contains is a very big set. We also embed into the set of all left invariant means on CB(G) which are mutually singular to every element of TLIM(CB(G)) for G = G1 ⨯ G2, where G1 is nondiscrete, non–compact, σ–compact and amenable as a discrete group and G2 is any amenable locally compact group. The extension of any left invariant mean on UCB(G) to CB(G) is discussed. We also provide an answer to a problem raised by Rosenblatt.
When is an invariant mean the limit of a Følner net?
