# Integration with respect to deficient topological measures on locally compact spaces

## TL;DR

This paper explores integration with respect to deficient topological measures on locally compact spaces, establishing properties, new measures, and convergence theorems, thus extending measure theory to a broader class of non-linear functionals.

## Contribution

It introduces a novel integration framework for deficient topological measures, characterizes resulting measures, and proves convergence results, expanding the theoretical understanding of non-linear measure-like functionals.

## Key findings

- Integration over sets yields new deficient topological measures.
- Such measures are absolutely continuous and Lipschitz continuous.
- Monotone convergence theorem is established.

## Abstract

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative vanishing at infinity function; and it produces a signed deficient topological measure if we use a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure and Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure $ \mu$ that assumes finitely many values, there is a function $ f $ such that $\int_X f \, d \mu = 0$, but $\int_X (-f )\, d \mu \neq 0$. We present different criteria for $\int_X f \, d \mu = 0$. We also prove some convergence results, including a Monotone convergence theorem.