The Hake-McShane and Hake-Henstock-Kurzweil integrals over m-dimensional unbounded sets

TL;DR

This paper extends certain integrals for Banach space valued functions from bounded open sets to unbounded sets with measure zero boundary, providing new characterizations via locally defined integrals.

Contribution

It introduces extensions of Hake-McShane and Hake-Henstock-Kurzweil integrals to unbounded sets with measure zero boundary, with full descriptive characterizations.

Findings

Extended integrals to unbounded sets with measure zero boundary

Provided full descriptive characterizations of the new integrals

Connected the new integrals with locally McShane and Henstock-Kurzweil integrals

Abstract

In this paper, we extend the Hake-McShane and Hake-Henstock-Kurzweil integrals of Banach space valued functions from m-dimensional open and bounded sets to m-dimensional sets G such that |G \ Go| = 0. We will prove the full descriptive characterizations of new integrals in terms of the locally McShane and locally Henstock-Kurzweil integrals.