Lebesgue-Stieltjes Measure on Time Scales

Abstract

The theory of time scales was introduced by Stefan Hilger in his Ph. D. thesis in 1988, supervised Bernd Auldbach, in order to unify continuous and discrete analysis [5]. Measure theory oil time scales was first constructed by Guseinov [4], then further studies were made by Guseinov-Bohner [1], Cabada-Vivero [2] and Rzezuchowski [6]. In this article, we adapt the concept of Lebesgue-Stieltjes measure to time scales. We define Lebesgue-Stieltjes Delta and del-measures and by using these measures, we define ail integral adapted to time scales, specifically Lebesgue-Stieltjes Delta-integral. We also establish the relation between Lebesgue-Stieltjes measure and Lebesgue-Stieltjes Delta-measure, consequently between Lebesgue-Stieltjes integral and Lebesgue-Stieltjes Delta-integral.