On the relation between Denjoy--Khintchine and HK$_{r}$-integrals

TL;DR

This paper explores the relationship between HK$_r$-integrals and Denjoy--Khintchine integrals, showing that under certain conditions, HK$_r$-integrals are included in the classical integral but the inclusion is strict.

Contribution

It clarifies the connection between HK$_r$-integrals and classical integrals, providing a direct proof of the proper inclusion under continuity restrictions.

Findings

HK$_r$-integrals are located within the approximate Henstock--Kurzweil integral theory.

When restricted to continuous indefinite integrals, HK$_r$-integrals are included in the Denjoy--Khintchine integral.

The inclusion of HK$_r$-integrals in the Denjoy--Khintchine integral is proper, not an equality.

Abstract

We locate Musial\,\&\,Sagher's concept of HK$_r$-inte\-gration within the approximate Henstock--Kurzweil integral theory. If to restrict HK$_r$-integral by requirement that the indefinite HK$_r$-integral is {\em continuous}, then it is included even in the classical Denjoy--Khintchine integral. We provide a direct argument demonstrating that this inclusion is proper.