On the extension of a Rieman–Stieltjes integral

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.