On the Henstock-Kurzweil Integral of C [a; b] Space-valued Functions

Abstract

There have been many contributions to the study of integration for mappings, taking values in ordered spaces. Among the authors, we quote Rie˘can [8], Duchon and Rie˘can [5], Rie˘can and Vr´abelov´a [9]. Henstock-Kurzweil-type integral for Riesz spaces-valued functions, defined on an interval [a, b] ⊂ R, was studied in detail by Boccuto, Rie˘can and Vr´abelov´a [3]. In the book, they assumed that Riesz spaces are Dedekind complete, that is, every bounded above subset of Riesz spaces has a supremum. In this paper, we will construct the Henstock-Kurzweil integral of C[a, b] space-valued functions, where C[a, b] means the collection of all real-valued continuous functions defined on a closed interval [a, b]. Before, we show that C[a, b] as a Riesz space but it is not Dedekind complete. Some properties of elements of C[a, b] were studied by Bartle and Sherbert [2]. They mentioned some of its properties are bounded, it has an absolute maximum and an absolute minimum, it can be approximated uniformly by step functions, uniformly continuous, and Riemann integrable. A property of C[a, b] is not a complete Dedekind Riesz space. Further discussion of C[a, b] can be shown in classical Banach spaces such as Albiac and Kalton [1], Diestel [4], Lindenstrauss and Tzafriri [6], Meyer-Nieberg [7], and others.