The Ross-Darboux-Stieltjes Integral

TL;DR

This paper introduces a generalized Darboux-Stieltjes integral that extends previous definitions, enlarges the class of integrable functions, and aligns with Lebesgue-Stieltjes, addressing limitations of traditional integrals.

Contribution

It presents a new generalized Darboux-Stieltjes integral consistent with earlier work, but with a broader class of integrable functions and improved properties.

Findings

The new integral agrees with all previous definitions.

It supports the Bounded Convergence Theorem.

It is equivalent to the Lebesgue-Stieltjes integral.

Abstract

Motivated by the limitations of the traditional definitions of the Riemann-Stieltjes and Darboux-Stieltjes integrals, we introduce a generalized Darboux-Stieltjes integral that is equivalent to an earlier generalization by Ross \cite{Ross}. Our definition builds upon an approach to the Darboux-Stieltjes integral recently introduced by the first author and Convertito \cite{TSI}. We show that our definition agrees with all previous definitions, but that the class of integrable functions is much larger. We develop all the analogs of the classic results for the Riemann integral, and rectify the problems inherent in the definition of the Darboux-Stieltjes integral in \cite{TSI}. In particular, we show the Bounded Convergence Theorem holds for our definition and that it agrees with the Lebesgue-Stieltjes integral.