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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2019, Volume 29, Issue 2, Pages 135–152 (Mi vuu672)

MATHEMATICS

On the extension of a Rieman–Stieltjes integral

V. Ya. Derr

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: In this paper, the properties of the regular functions and the so-called $\sigma$-continuous functions (i.e., the bounded functions for which the set of discontinuity points is at most countable) are studied. It is shown that the $\sigma$-continuous functions are Riemann–Stieltjes integrable with respect to continuous functions of bounded variation. Helly's limit theorem for such functions is also proved. Moreover, Riemann–Stieltjes integration of $\sigma$-continuous functions with respect to arbitrary functions of bounded variation is considered. To this end, a $(*)$-integral is introduced. This integral consists of two terms: (i) the classical Riemann–Stieltjes integral with respect to the continuous part of a function of bounded variation, and (ii) the sum of the products of an integrand by the jumps of an integrator. In other words, the $(*)$-integral makes it possible to consider a Riemann–Stieltjes integral with a discontinuous function as an integrand or an integrator. The properties of the (*)-integral are studied. In particular, a formula for integration by parts, an inversion of the order of the integration theorem, and all limit theorems necessary in applications, including a limit theorem of Helly's type, are proved.

Keywords: functions of bounded variation, regulated functions, $\sigma$-continuous functions, Rieman–Stieltjes integral, $(*)$-integral.

DOI: https://doi.org/10.20537/vm190201

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Bibliographic databases:

UDC: 517.518.126
MSC: 26B30, 26A42

Citation: V. Ya. Derr, “On the extension of a Rieman–Stieltjes integral”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:2 (2019), 135–152

Citation in format AMSBIB
\Bibitem{Der19} \by V.~Ya.~Derr \paper On the extension of a Rieman--Stieltjes integral \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2019 \vol 29 \issue 2 \pages 135--152 \mathnet{http://mi.mathnet.ru/vuu672} \crossref{https://doi.org/10.20537/vm190201} \elib{https://elibrary.ru/item.asp?id=39136239} 

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This publication is cited in the following articles:
1. Fedorov D.L., Gryzlov A.A., Kinzebulatov D.M., Latypova N.V., Maksimov V.P., Petrov N.N., Popova S.N., Rodionov V.I., Rodina L.I., Smetanin Yu.M., Zaitsev V.A., “Vasilii Yakovlevich Derr. to Anniversary”, Vestn. Udmurt. Univ.-Mat. Mekh. Kompyuternye Nauk., 29:4 (2019), 612–617
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