Some Elementary Properties of Kurzweil-Henstock-Stieltjes Integral on \(\mathbb{R}^n\)
Justine Bryle C. Macaso *
Department of Mathematics, Central Mindanao University, Philippines.
Greig Bates C. Flores
Mathematics Faculty, Department of Mathematics, Central Mindanao University, Philippines.
*Author to whom correspondence should be addressed.
Abstract
Kurzweil-Henstock integral is a generalization of the Reimann integral. In this paper, we established the definition of Kurzweil-Henstock-Stieltjes integral on \(\mathbb{R}\)n via gauge type approach where integrand and integrator are all real-valued functions defined on a compact interval in \(\mathbb{R}\)n. Moreover, the Cauchy Criterion is established. To this end, some underlying simple properties of this integral are studied, specifically, uniqueness, linearity, monotonocity, integrability over a subset, and additivity. Results gathered in this paper may serve as a foundation to some related studies such as the notion of convergence with respect to this integral, and the formulation of the Saks-Henstock Lemma.
Keywords: Stieltjes, perron partition, \(\sigma\)-fine, cauchy criterion