Convergence theorems for several decomposition type non-linear integrals

TL;DR

This paper classifies and compares various non-linear integrals based on approximation direction, set family types, coefficient signatures, and the number of terms, focusing on their convergence properties.

Contribution

It provides a comprehensive classification and comparison of convergence theorems for different types of non-linear integrals.

Findings

Identifies conditions for monotone convergence from above and below

Classifies integrals by set family types and coefficient signatures

Establishes comparison results among various non-linear integrals

Abstract

These are classified by the direction of approximation (from above or below), the set family types (partition or covering) of simple functions, the coefficient signature (non-negative or signed), and cardinal number of terms of simple functions(finite or countable infinite). We will compare these integrals considering the monotone increasing/decreasing convergence theorems.