# Properties of the Riemann-Lebesgue integrability in the non-additive case

## TL;DR

This paper explores the properties of Riemann-Lebesgue integrability for vector functions with respect to non-additive set functions, establishing classical properties and relationships among different integrability concepts.

## Contribution

It extends classical integral properties to the non-additive case and clarifies the relationships among Riemann-Lebesgue, Birkhoff simple, and Gould integrabilities.

## Key findings

- Classical integral properties are valid in the non-additive setting.
- Continuity properties of the integral are characterized.
- Relationships among various integrability notions are established.

## Abstract

We study Riemann-Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann-Lebesgue, Birkhoff simple and Gould integrabilities are also established.