Signed Topological Measures on Locally Compact Spaces

Abstract

In this paper we define and study signed deficient topological measures and signed topological measures (which generalize signed measures) on locally compact spaces. We prove that a signed deficient topological measure is τ -smooth on open sets and τ -smooth on compact sets. We show that the family of signed measures that are differences of two Radon measures is properly contained in the family of signed topological measures, which in turn is properly contained in the family of signed deficient topological measures. Extending known results for compact spaces, we prove that a signed topological measure is the difference of its positive and negative variations if at least one variation is finite; we also show that the total variation is the sum of the positive and negative variations. If the space is locally compact, connected, locally connected, and has the Alexandroff one-point compactification of genus 0, a signed topological measure of finite norm can be represented as a difference of two topological measures.2010 Mathematics Subject Classification. Primary 28C15; Secondary 28C99. Key words and phrases. signed deficient topological measure; signed topological measure; positive, negative, total variation of a signed deficient topological measure.

Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces

“…When X is compact, there are examples of topological measures that are not measures and of deficient topological measures that are not topological measures in numerous papers, beginning with [1], [10], and [14]. When X is locally compact, see [2], [4,Sections 5 and 6], and [3, Section 9] for more information on proper inclusion, criteria for a deficient topological measure to belong to M (X) or T M (X) (in particular, to be a Radon measure or a regular Borel measure), as well as various examples.…”