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

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.

Received: 18.03.2019

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

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