Integration by Parts and Other Theorems for R3S-Integrals

Abstract

This paper is a continuation of [3], in which was introduced the Refinement-Ross-Riemann-Stieltjes $(R^3S)$ Integral, and in which some of its advantages were exhibited. After a brief summary of [3], this paper proves an integration by parts theorem which shows incidentally that if $f$ is $R^3S$-integrable with respect to $g$ then $g$ is $R^3S$-integrable with respect to $f$. Theorems on term-by-term integration of sequences analogous to the Helly-Bray Theorem are next proved, in a context of Wiener's functions of bounded generalized variation as developed by L. C. Young and me. In a similar context I prove also a theorem resembling the classical theorem of Riesz representing linear functionals by Stieltje.