Turkish Journal of Mathematics

Abstract: The theory of time scales was introduced by Stefan Hilger in his Ph. D. thesis in 1988, supervised by Bernd Auldbach, in order to unify continuous and discrete analysis [5]. Measure theory on 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 D and \nabla-measures and by using these measures, we define an integral adapted to time scales, specifically Lebesgue-Stieltjes D-integral. We also establish the relation between Lebesgue-Stieltjes measure and Lebesgue-Stieltjes D-measure, consequently between Lebesgue-Stieltjes integral and Lebesgue-Stieltjes D- integral.

Key Words: Time scales, Lebesgue-Stieltjes D-measure, Lebesgue-Stieltjes D-integral.