Covering properties and F�lner conditions for locally compact groups

Abstract

LetG be a locally compact group with left Haar measurem G on the Borel sets IB(G) (generated by open subsets) and write |E|=m G (E). Consider the following geometric conditions on the groupG. (FC If ɛ>0 and compact setK⊂G are given, there is a compact setU with 0 0 and compact setK⊂G, which includes the unit, are given there is a compact setU with 0<|U|<∞ and |K U ΔU|/|U|<ɛ. HereA ΔB=(A/B)⌣(B/A) is the symmetric difference set; by regularity ofm G it makes no difference if we allowU 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” ofG: the existence of a left invariant mean on the spaceCB(G) of all continuous bounded functions. We show, for arbitrary locally compact groupsG, that (amenable)⇔(FC)⇔(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact groupG satisfies. (C) For at least one setK, with int(K)≠O and $$\bar K$$ compact, there is an indexed family {x α∶αeJ}⊂G such that {Kx α} is a covering forG whose covering index at each pointg (the number of αeJ withgeKx α) is uniformly bounded throughoutG.