Invariant means on the continuous bounded functions

Abstract

Let G be a noncompact nondiscrete σ \sigma -compact locally compact metric group. A Baire category argument gives measurable sets { A γ : γ ∈ Γ } \{ {A_\gamma }:\gamma \in \Gamma \} of finite measure with card ( Γ ) = c (\Gamma ) = c which are independent on the open sets. One approximates { A γ : γ ∈ Γ } \{ {A_\gamma }:\gamma \in \Gamma \} by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions { f γ : γ ∈ Γ } \{ {f_\gamma }:\gamma \in \Gamma \} in CB ( G ) {\text {CB}}(G) with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on CB ( G ) {\text {CB}}(G) there are 2 c {2^c} mutually singular elements of LIM each of which is singular to TLIM.