# Deficient topological measures on locally compact spaces

## TL;DR

This paper introduces and studies deficient topological measures on locally compact spaces, exploring their properties, variations, and conditions under which they become measures or topological measures, thereby extending the theory of measures.

## Contribution

It defines deficient topological measures, investigates their properties, and provides criteria for when they are measures or topological measures, advancing the understanding of non-linear functionals.

## Key findings

- Deficient topological measures generalize measures and topological measures.
- Positive, negative, and total variations of signed set functions are deficient topological measures.
- Conditions are established for deficient topological measures to be measures or topological measures.

## Abstract

Topological measures and quasi-linear functionals generalize measures and linear functionals. We define and study deficient topological measures on locally compact spaces. A deficient topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on compact sets, inner regular on open sets, and outer regular on closed sets. Deficient topological measures generalize measures and topological measures. First we investigate positive, negative, and total variation of a signed set function that is only assumed to be finitely additive on compact sets. These positive, negative, and total variations turn out to be deficient topological measures. Then we examine finite additivity, superadditivity, smoothness, and other properties of deficient topological measures. We obtain methods for generating new deficient topological measures. We provide necessary and sufficient conditions for a deficient topological measure to be a topological measure and to be a measure. The results presented are necessary for further study of topological measures, deficient topological measures, and corresponding non-linear functionals on locally compact spaces.