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Home » Issues » Archive of Issues » 2019 - 2 issue » pp. 135-152

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Russia Izhevsk
Year
2019
Volume
29
Issue
2
Pages
135-152
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Section Mathematics
Title On the extension of a Rieman-Stieltjes integral
Author(-s) Derr V.Ya.a
Affiliations Udmurt State Universitya
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
UDC 517.518.126
MSC 26B30, 26A42
DOI 10.20537/vm190201
Received 18 March 2019
Language Russian
Citation Derr V.Ya. On the extension of a Rieman-Stieltjes integral, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 135-152.
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