# Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces

## TL;DR

This paper explores the relationships between various non-linear functionals and deficient topological measures on locally compact spaces, establishing representation theorems and bijections that unify different classes of measures and functionals.

## Contribution

It introduces new representation theorems and bijections linking non-linear functionals with deficient topological measures on locally compact spaces.

## Key findings

- Established an order-preserving, conic-linear bijection between finite deficient topological measures and bounded p-conic quasi-linear functionals.
- Proved representation theorems connecting non-linear functionals with deficient topological measures.
- Provided equivalent definitions of quasi-linear functionals and functionals for deficient topological measures in compact spaces.

## Abstract

We study non-linear functionals, including quasi-linear functionals, p-conic quasi-linear functionals, d-functionals, r-functionals, and their relationships to deficient topological measures and topological measures on locally compact spaces. We prove representation theorems and show, in particular, that there is an order-preserving, conic-linear bijection between the class of finite deficient topological measures and the class of bounded p-conic quasi-linear functionals. Our results imply known representation theorems for finite topological measures and deficient topological measures. When the space is compact we obtain four equivalent definitions of a quasi-linear functional and four equivalent definitions of functionals corresponding to deficient topological measures.