Deficient topological measures on locally compact spaces

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 sets 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.

Automated summary

This article investigates deficient topological measures on locally compact spaces, exploring properties such as positive, negative, and total variation, as well as finite additivity. It presents methods for generating new deficient topological measures and provides necessary and sufficient conditions for a deficient topological measure to be a topological measure or a measure. These results are crucial for further research on topological measures, deficient topological measures, and corresponding non-linear functionals on locally compact spaces.