Covering properties and Følner conditions for locally compact groups

W. R. Emerson, F. P. Greenleaf

Research output: Contribution to journalArticlepeer-review

Abstract

Let G be a locally compact group with left Haar measure mG on the Borel sets IB(G) (generated by open subsets) and write |E|=mG(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 mG 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 {xα:αεJ}⊂G such that {Kxα} is a covering for G whose covering index at each point g (the number of αεJ with gεKxα) is uniformly bounded throughout G.

Original languageEnglish (US)
Pages (from-to)370-384
Number of pages15
JournalMathematische Zeitschrift
Volume102
Issue number5
DOIs
StatePublished - Oct 1967

ASJC Scopus subject areas

  • Mathematics(all)

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