## 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

Zbl 1416.26013
Full Text:

### References:

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