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On a generalization of Henstock-Kurzweil integrals. (English) Zbl 1474.26033

The paper introduces a new class of Stieltjes-type gauge integrals called the \(\mbox{HKS}_\alpha^p\) integrals. Given an open interval \(I\subset\mathbb{R}\), a measurable function \(F:I\to\mathbb{R}\) is called an indefinite \(\mbox{HKS}_\alpha^p\) integral of a measurable function \(f:I\to\mathbb{R}\) with respect to a measurable function \(G:I\to\mathbb{R}\), provided that for each \(\varepsilon>0\), there exists a gauge \(\delta:I\to(0,\infty)\) such that the inequality \[\sum_{i=1}^m\mbox{osc}_p(F-f(x_i)G,[a_i,b_i])<\varepsilon\] holds for each \(\delta\)-fine \(\alpha\)-partition \(([a_i,b_i],x_i)_{i=1}^m\) in \(I\). It is assumed that \(\alpha\ge 1\), and an \(\alpha\)-partition in \(I\) is a collection of intervals such that \((x_i-\alpha(x_i-a_i),x_i+\alpha(b_i-x_i))\) are pairwise disjoint intervals in \(I\). Moreover, the symbol \(\mbox{osc}_p(h,[a,b])\) stands for the \(p\)-oscillation of a measurable function \(h\) on \([a,b]\); a precise definition is given in the paper.
Centered \(\mbox{HKS}_\alpha^p\) integrals are defined in a similar way, but it is assumed that each \(x_i\) is the center of the corresponding interval \([a_i,b_i]\).
Basic properties of \(\mbox{HKS}_\alpha^p\) integrals are established. It is shown that the classical Henstock-Kurzweil-Stieltjes integral, as well as the more recent \(\mbox{MC}_\alpha\) integral (see [T. Ball and D. Preis, in: Mathematics almost everywhere. Hackensack, NJ: World Scientific. 69–92 (2018; Zbl 1416.26013)]), are special cases of the \(\mbox{HKS}_\alpha^p\) integral. The authors also investigate the dependence of the class of \(\mbox{HKS}_\alpha^p\) integrable functions on the choice of \(p\), the relation between \(\mbox{HKS}_\alpha^p\) integrals and the Denjoy-Khinchin integral, and other topics.
This landmark paper will be of interest to all researchers in integration theory.

MSC:

26A39 Denjoy and Perron integrals, other special integrals

Citations:

Zbl 1416.26013
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References:

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