# Repeated quasi-integration on locally compact spaces

## TL;DR

This paper investigates the properties of quasi-integrals on locally compact spaces, providing criteria for their repeated integration to produce quasi-linear functionals and characterizing when different orders of integration yield the same results.

## Contribution

It introduces criteria for repeated quasi-integration to produce quasi-linear functionals and characterizes when different integration orders are equivalent in this context.

## Key findings

- Criteria for repeated quasi-integration to yield quasi-linear functionals
- Conditions under which double quasi-integrals are simple
- Characterization of when order of integration does not affect the result

## Abstract

When $X$ is locally compact, a quasi-integral (also called a quasi-linear functional) on $ C_c(X)$ is a homogeneous, positive functional that is only assumed to be linear on singly-generated subalgebras. We study simple and almost simple quasi-integrals, i.e., quasi-integrals whose corresponding compact-finite topological measures assume exactly two values. We present a criterion for repeated quasi-integration (i.e., iterated integration with respect to topological measures) to yield a quasi-linear functional. We find a criterion for a double quasi-integral to be simple. We describe how a product of topological measures acts on open and compact sets. We show that different orders of integration in repeated quasi-integrals give the same quasi-integral if and only if the corresponding topological measures are both measures or one of the corresponding topological measures is a positive scalar multiple of a point mass.