# Decompositions of signed deficient topological measures

## TL;DR

This paper explores the decomposition of various generalized set functions like topological and deficient measures, extending known results to locally compact spaces and introducing proper signed deficient measures.

## Contribution

It introduces the concept of proper signed deficient topological measures and shows their decomposition into signed Radon measures plus proper measures.

## Key findings

- Signed deficient topological measures can be decomposed into a signed Radon measure and a proper signed deficient measure.
- The sum of two proper (deficient) topological measures remains proper (deficient).
- Criteria are provided for when a (deficient) topological measure is proper.

## 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 $\infty$ or $-\infty$. 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.