
Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2019, Volume 29, Issue 2, Pages 135–152
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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 socalled $\sigma$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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UDC:
517.518.126
MSC: 26B30, 26A42 Received: 18.03.2019
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
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\by V.~Ya.~Derr
\paper On the extension of a RiemanStieltjes integral
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2019
\vol 29
\issue 2
\pages 135152
\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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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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