Some remarks on characterization of t-normed integrals on compacta

TL;DR

This paper discusses the characterization of t-normed integrals on compact spaces, establishing a correspondence between capacities and certain monotone functionals, extending classical measure-theoretic results to a non-linear setting.

Contribution

It extends previous characterizations of t-normed integrals to general compacta and discusses the optimality of these characterizations.

Findings

Established a correspondence between capacities and t-norm monotone functionals.

Extended characterization from finite to general compacta.

Analyzed the optimality of the existing characterization.

Abstract

A characterization of t-normed integrals was obtained in \cite{CLM} for finite compacta and in \cite{Rad} for the general case. Such characterization establishes a correspondence between the space of capacities and homogeneous respect t-norm monotone normed functionals preserving the maximum operation of comonotone functions. In fact these theorems we can consider as non-additive and non-linear analogues of well-known Riesz Theorem about a correspondence between the set of $\sigma$-additive regular Borel measures and the set of linear positively defined functionals. We discuss optimality of such characterization.