Dunford-Henstock-Kurzweil and Dunford-Mcshane integrals of vector-valued functions defined on m-dimensional bounded sets

TL;DR

This paper introduces new vector-valued integrals extending classical integrals to m-dimensional measurable sets, providing characterizations and relationships with existing integrals.

Contribution

It defines Dunford-Henstock-Kurzweil and Dunford-McShane integrals for functions on bounded measurable sets, extending classical integrals to higher dimensions.

Findings

New integrals are natural extensions of classical integrals.

Full descriptive characterizations of McShane and Henstock-Kurzweil integrals.

Relationships established between the new integrals and the Dunford integral.

Abstract

In this paper, we define the Dunford-Henstock-Kurzweil and the Dunford-McShane integrals of Banach space valued functions defined on a bounded Lebesgue measurable subset of m-dimensional Euclidean space Rm. We will show that the new integrals are "natural" extensions of the McShane and the Henstock-Kurzweil integrals from m-dimensional closed non-degenerate intervals to m-dimensional bounded Lebesgue measurable sets. As applications, we will present full descriptive characterizations of the McShane and Henstock-Kurzweil integrals in terms of our integrals. Moreover, a relationship between new integrals will be proved in terms of the Dunford integral.