Measure Theory on Times Scales

Abstract

In this thesis, we have studied measure theory adapted to time scales. delta and nabla-measures were first defined by Guseinov in 2003, then in a further study, the relationship between Lebesgue delta-integral and Riemann delta-integral were introduced in detail by Guseinov and Bohner. In 2004, Cabada established the relationship between delta-measure and the classical Lebesgue measure, moreover, Lebesgue delta-integral and the classical Lebegue integral. Finally, deltameasurability of sets was studied by Rzezuchovsky in 2005. In this study, we have adapted basic concepts of the measure theory to time Scales, by using definitions and properties given in these papers. With the help of related papers, Lebesgue-Stieltjes measure has been constructed on time scales and the link between Lebesgue-Stieltjes measure and Lebesgue-Stieltjes delta-measure and also link between Lebesgue-Stieltjes delta-integral and Lebesgue-Stieltjes integral have taken place.