Integral Henstock-Kurzweil di Dalam Ruang C[Alfa,Beta]

The research entitled “Integral Henstock-Kurzweil di Dalam Ruang C[Alfa,Beta]” was conducted by Firdaus Ubaidillah under the guidance of Prof. Dr. Soeparna Darmawijaya and Prof. Dr. Ch. Rini Indrati in 2017.

The following is the abstract of this research.

ABSTRACT

This dissertation is the result of research to construct of Henstock-Kurzweil type integral for $C[\alpha,\beta]$ space-valued functions that defined on a closed interval $[f,g]\subseteq C [\alpha, \beta]$ , where $C[\alpha,\beta]$ is the collection of all real-valued continuous functions defined on a closed interval $[\alpha,\beta]\subseteq \mathbb$ . The concept of this integral follows Henstock-Kurzweil integral concept in the real space that has been known so far. We begin by introducing some fundamental concepts about the space $C [\alpha, \beta]$ , such as some properties of $C [\alpha, \beta]$ as a Riesz space, convergence of sequences and series, $C[\alpha,\beta]$ space-valued norms and metrics, and neighborhood of a point in $C [\alpha, \beta]$ . Furthermore, we construct calculus on $C [\alpha, \beta]$ for constructing Henstock-Kurzweil integral in the space $C [\alpha, \beta]$ such as limit, continuity, derivative of $C [\alpha , \beta]$ space-valued functions, and convergence of a sequence of functions. To construc the Henstock-Kurzweil integral of a $C [\alpha, \beta]$ space-valued function that defined on a closed interval $[f, g] \subseteq C [\alpha, \beta]$ , we begin by constructing a partition on $[f, g]$ . Furthermore, we define Henstock-Kurzweil integral of a $C [\alpha, \beta]$ space-valued function that defined on a closed interval $[f, g]\subseteq C[\alpha,\beta]$ . From this definition, we develop some properties of a Henstock-Kurzweil integrable function into theorems. From a Henstock-Kurzweil integrable function, we define a primitive of a Henstock-Kurzweil integrable function. To know some properties of a primitive of a Kurzweil-Henstock integrable function, we discuss the absolute Henstock-Kurzweil integral. In this dissertation, we give a brief discussion to Denjoy integral of $C [\alpha , \beta]$ space-valued functions. Further, we show that Denjoy integral is equivalent with Henstock-Kurzweil integral. To discuss this, we introduce the bounded variation function and the absolute continuous function. Finally, we discuss some convergence theorems of a sequence of Henstock-Kurzweil integrable functions. Our objective here is to prove that the monotone, uniform, controlled and dominated convergences of a sequence of Henstock-Kurzweil integral functions imply the convergence of the sequence formed by its corresponding integrals.

Kata Kunci: Henstock-Kurzweil integral, Henstock-Kurzweil primitive, bounded variation function, absolute continuous function, convergence theorem