Decompositions of signed deficient topological measures

Abstract

This paper focuses on various decompositions of topological measures, deficient topological measures, signed topological measures, and signed deficient topological measures. These set functions generalize measures and correspond to certain non-linear functionals. They may assume ∞ or −∞. We introduce the concept of a proper signed deficient topological measure and show that a signed deficient topological measure can be represented as a sum of a signed Radon measure and a proper signed deficient topological measure. We also generalize practically all known results that involve proper deficient topological measures and proper topological measures on compact spaces to locally compact spaces. We prove that the sum of two proper (deficient) topological measures is a proper (deficient) topological measure. We give a criterion for a (deficient) topological measure to be proper.

Weak convergence of topological measures

“…Then λ({x}) ≤ λ(X \ B) = λ(X) − λ(B) = 0. Thus, λ({x}) = 0 for any x ∈ X, and by [12,Lemma 4.15] λ is proper.…”

“…Remark 35. From [12,Theorem 4.3] it follows that a finite deficient topological measure can be written as a sum of a finite Radon measure and a proper finite deficient topological measure. The sum of two proper deficient topological measures is proper (see [12,Theorem 4.8]).…”

“…A finite Radon measure on a compact space is a regular Borel measure, so our definition (which is given in [12]) of a proper deficient topological measure coincides with definitions in papers prior to [12].…”

“…By Remark 35 µ N is a proper deficient topological measure. By [12,Theorem 4.5] there are compact sets K 1 , . .…”

“…It is easy to see that ν i is a deficient topological measure, and ν i (X) = λ i (X i ) < ∞. Since λ i is proper, by [12,Theorem 4.5] given δ > 0 there are sets of the form V j ∩ X i , V j ∈ O(X), j = 1, . .…”