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Lebesgue-Stieltjes measure on time scales
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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 . Measure theory on time scales was first constructed by Guseinov , then further studies were made by Guseinov-Bohner , Cabada-Vivero  and Rzezuchowski . In this article, we adapt the concept of Lebesgue-Stieltjes measure to time scales. We define Lebesgue-Stieltjes Δ and ▶-measures and by using these measures, we define an integral adapted to time scales, specifically Lebesgue-Stieltjes Δ-integral. We also establish the relation between Lebesgue-Stieltjes measure and Lebesgue-Stieltjes Δ-measure, consequently between Lebesgue-Stieltjes integral and Lebesgue-Stieltjes Δ-integral.