Covering properties and Følner conditions for locally compact groups

Let G be a locally compact group with left Haar measure m_G on the Borel sets IB(G) (generated by open subsets) and write |E|=m_G(E). Consider the following geometric conditions on the group G. (FC If e{open}>0 and compact set K⊂G are given, there is a compact set U with 0<|U|<∞ and |x U ΔU|/|U|<e{open} for all xεK. (A) If e{open}>0 and compact set K⊂G, which includes the unit, are given there is a compact set U with 0<|U|<∞ and |K U ΔU|/|U|<e{open}. Here A ΔB=(A/B){smile}(B/A) is the symmetric difference set; by regularity of m_G it makes no difference if we allow U to be a Borel set. It is well known that (A)⇒(FC) and it is known that validity of these conditions is intimately connected with "amenability" of G: the existence of a left invariant mean on the space CB(G) of all continuous bounded functions. We show, for arbitrary locally compact groups G, that (amenable)⇔(FC)⇔(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact group G satisfies. (C) For at least one set K, with int(K)≠Ø and {Mathematical expression} compact, there is an indexed family {x_α:αεJ}⊂G such that {Kx_α} is a covering for G whose covering index at each point g (the number of αεJ with gεKx_α) is uniformly bounded throughout G.