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 RiemannStieltjes integrable with respect to continuous functions of bounded variation. Helly's limit theorem for such functions is also proved. Moreover, RiemannStieltjes 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 RiemannStieltjes 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 RiemannStieltjes 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.

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