Radon measures on groups

Abstract

Let G be a Hausdorff topological group. We shall be dealing with positive and real valued Radon measures [3] on G. Following [1], a real valued Radon measure ,u on G is said to be mobile if, for every compact set KCG, the function a--+1A(aK) is continuous on G. In the case of a locally compact Hausdorff group any bounded mobile Radon measure is the indefinite integral (with respect to Haar measure) of a function in L'(G) [1 ]. We shall prove in this note that if there exists a (nontrivial) mobile real (not necessarily bounded) Radon measure on G, then the group is locally compact (Theorem 2). An immediate consequence is the following converse to the existence of Haar measure. If in a Hausdorff topological group there is a nontrivial ( g 0) left translation invariant Radon measure then the group is locally compact. Let us first recall DEFINITION 1. A Radon measure ,u on a Hausdorff topological space X is a positive measure defined on the Borel subsets of X satisfying (1) IA is locally finite and (2) IA is inner regular i.e. for every Borel set B, IA(B) =sup {II(K): compact K CB }. A real valued Radon measure v is a signed Borel measure such that v+ and vare both Radon (equivalently v is the difference of two positive Radon measures, one of which (at least) is totally finite). We shall need the following fact about Radon measures. For every compact set K and any E> 0, there exists an open set VD K such that IA(V) <I(K) + e