Role of the Harnack Extension Principle in the Kurzweil-Stieltjes Integral

TL;DR

This paper explores the Harnack extension principle's role in Kurzweil-Stieltjes integrals, especially with discontinuous integrators, introducing new concepts to extend convergence theorems and ensure integral existence over arbitrary subsets.

Contribution

It develops a generalized Harnack extension principle for Kurzweil-Stieltjes integrals with discontinuous integrators, incorporating new concepts like equi-integrability and equiregulatedness.

Findings

Harnack extension principle is pivotal for convergence theorems.

New concepts enable extension of the principle to discontinuous integrators.

Ensures existence of integrals over arbitrary subsets of the domain.

Abstract

Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence theorems provide one of the most widely used tools. The Harnack extension principle, which discusses a sufficient condition for Kurzweil-Henstock integrable functions on particular subsets of $(a,b)$ to be integrable on $[a,b]$, is a key step to supply convergence theorems. The Kurzweil-Stieltjes integral reduces to the Kurzweil-Henstock integral whenever the integrator is an identity function. In general, if the integrator $F$ is discontinuous on $[c,d]\subset[a,b]$, then the values of the Kurzweil-Stieltjes integrals $$\int_c^d[dF]g,\ \int_{[c,d]}[dF]g,\ \int_{[c,d)}[dF]g,\ \int_{(c,d]}[dF]g,\ {\rm and}\ \int_{(c,d)}[dF]g$$ need not coincide. Hence, the Harnack extension principle in the Kurzweil-Henstock integral cannot be valid any longer for the Kurzweil-type Stieltjes integrals with discontinuous integrators. The new concepts of equi-integrability and equiregulatedness are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration. Moreover, the existence of the integral $\int_a^b[dF]g$ does not (even in the case of the identity integrator) always imply the existence of the integral $\int_{T}[dF]g$ for every subset $T$ of $[a,b]$. This follows from the well-known fact that, if e.g., $T\subset[a,b]$ is not measurable, then the existence of the Lebesgue integral $\int_a^b g [dt]$ does not imply that the integral $\int_T g [dt]$ exists. Therefore, besides constructing the Harnack extension principle for the abstract Kurzweil-Stieltjes integral, the aim of this paper is also to demonstrate its role in guaranteeing the existence of the integrals $\int_{T}[dF]g$ for arbitrary subsets $T$ of an elementary set $E$.