Role of the Harnack extension principle in the Kurzweil-Stieltjes integral
Umi Mahnuna Hanung
Received November 29, 2022. Published online July 17, 2023.
Abstract: In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with discontinuous integrators. Moreover, in general, the existence of integral over an elementary set $E$ does not always imply the existence of integral over every subset $T$ of $E.$ The goal of this paper is to construct the Harnack extension principle for the Kurzweil-Stieltjes integral with values in Banach spaces and then to demonstrate its role in guaranteeing the integrability over arbitrary subsets of elementary sets. New concepts of equiintegrability and equiregulatedness involving elementary sets are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration.
Keywords: Kurzweil-Stieltjes integral; integral over arbitrary bounded sets; equiintegrability; equiregulatedness; convergence theorem; Harnack extension principle
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Affiliations: Umi Mahnuna Hanung, Korteweg de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands; Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha, Czech Republic; and Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia, e-mail: hanungum@ugm.ac.id, H.UmiMahnuna@uva.nl
