Adjoint classes of Kurzweil–Stieltjes integrable functions

Abstract

It is well known that functions of bounded variation, BV[a, b], and continuous functions, C[a, b], define two classes of functions which are adjoint with respect to the Riemann–Stieltjes integral. This means that: (1) if the integral $$\int _a^b f\,dg$$ exists for every function g in BV[a, b], then f belongs to C[a, b]; (2) if the integral $$\int _a^b f\,dg$$ exists for every function f in C[a, b], then g belongs to BV[a, b]. In this paper, we analyse whether the Kurzweil–Stieltjes integral mimics such a property of the Riemann–Stieltjes integral with respect to some classes of functions. In particular, we investigate the validity of (1) and (2) in the case when the integral is the Kurzweil–Stieltjes integral and C[a, b] is replaced by the class of regulated functions G[a, b] (i.e., functions with discontinuities of the first kind).