On the approximately continuous Foran integral: completing our chart (Q930102)

This paper is a continuation of the previous work of this author [Real Anal. Exch. 30, No. 1, 183--192 (2004-2005; Zbl 1060.26011)]. The main result of this article is the following theorem: Let a function \(f:[a,b]\to\mathbb R\) be \({\mathcal F}_4\) - and \(AF\) - integrable (the \({\mathcal F}_4\) - integral is \(AK_N\) - integral of Gordon and \(AF\) - integral denotes the approximate Foran integral). Then both values are equal. However, the author indicates also that for \({\mathcal F}_4\) - integral and the approximate Foran integral there are functions integrable in one sense but not in the other.