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

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.