Choquet-Sugeno-like operator based on relation and conditional aggregation operators

TL;DR

This paper introduces a versatile Choquet-Sugeno-like operator that generalizes many existing operators for bounded functions and monotone measures, based on dependence relations and conditional aggregation, without relying on $t$-level sets.

Contribution

The paper proposes a new unified operator framework that encompasses various known integrals and operators, expanding the theoretical understanding of aggregation in measure theory.

Findings

The operator generalizes several existing measures and integrals.

Conditions are provided for the operator to coincide with known integrals.

Basic properties of the new operator are established.

Abstract

We introduce a~\textit{Choquet-Sugeno-like operator} generalizing many operators for bounded functions and monotone measures from the literature, e.g., Sugeno-like operator, Lov\'{a}sz and Owen measure extensions, $\rF$-decomposition integral with respect to a~partition decomposition system, and others. The new operator is based on the concepts of dependence relation and conditional aggregation operators, but it does not depend on $t$-level sets. We also provide conditions for which the Choquet-Sugeno-like operator coincides with some Choquet-like integrals defined on finite spaces and appeared recently in the literature, e.g. reverse Choquet integral, $d$-Choquet integral, $\rF$-based discrete Choquet-like integral, some version of $C_{\rF_1\rF_2}$-integral, $\mathrm{C}\mathrm{C}$-integrals (or Choquet-like Copula-based integral) and discrete inclusion-exclusion integral. Some basic properties of the Choquet-Sugeno-like operator are studied.