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metric group. A Baire category argument gives measurable sets (AT: y eI r) of finite measure with card(I) c which are independent on the open sets. One approximates (AT: y E r} by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions {(f: y E r) in CB(G) with a cetain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on CB(G) there are 2" mutually singular elements of LIM each of which is singular to TLIM.

Invariant means on the continuous bounded functions