A generalized Henstock-Stieltjes integral involving division functions. (English) Zbl 1174.26006
The Riemann-Stieltjes integral is well-known. In the integral of \(f\) with respect to \(g\), we may regard \(f\) as a point function and \(g\) as an interval function. In a series of papers, the authors extended the integral to an integral in which \(f\) is a point function and \(g\) is a division function, where division is defined as in the Henstock integral [P. Y. Lee and V. Vyborny, The Integral: An easy Approach after Kurzweil and Henstock. Australian Mathematical Society Lecture Series. 14. Cambridge: Cambridge University Press (2000; Zbl 0941.26003)].
In this paper, the authors further improve the integral so that a full theory of the integral can be developed. The integral is called the generalized Henstock-Stieltjes integral or the \(GS_k\) integral. The authors use two functions in the definition of the integral and impose other conditions so that they can prove the standard results in the theory of integration, including the Saks-Henstock lemma, integration by parts, and the controlled convergence theorem, not given in the previous papers. Examples of such integrals are also given.
In this paper, the authors further improve the integral so that a full theory of the integral can be developed. The integral is called the generalized Henstock-Stieltjes integral or the \(GS_k\) integral. The authors use two functions in the definition of the integral and impose other conditions so that they can prove the standard results in the theory of integration, including the Saks-Henstock lemma, integration by parts, and the controlled convergence theorem, not given in the previous papers. Examples of such integrals are also given.
Reviewer: Rudolf Výborný (Queensland)
MSC:
| 26A39 | Denjoy and Perron integrals, other special integrals |
Keywords:
Henstock integral; \(\delta ^k\)-fine division; Saks-Henstock lemma; \(GR_k\)-integral; \(GR_{k}^{*}\)-integral; weakly \(g\)-regular function; \(SV^{k}[a,b]\); \(g\)-nearly additive functionCitations:
Zbl 0941.26003References:
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