Multiple Lebesgue Integration on Time Scales

Differential and integral calculus on time scales allows to develop a theory of dynamic equations in order to unify and extend the usual differential equations and difference equations. For single variable differential and integral calculus on time scales, we refer the reader to the textbooks [4, 5] and the references given therein. Multivariable calculus on time scales was developed by the authors [2, 3]. In [3], we presented the process of Riemann multiple delta (nabla and mixed types) integration on time scales. In the present paper, we introduce the definitions of Lebesgue multi-dimensional delta (nabla and mixed types) measures and integrals on time scales. A comparison of the Lebesgue multiple delta integral with the Riemann multiple delta integral is given. Beside this introductory section, this paper consists of two sections. In Section 2, following [3], we give the Darboux definition of the Riemann multiple delta integral and present some needed facts connected to it. Themain part of this paper is Section 3. There, a brief description of the Carathéodory construction of a Lebesgue measure in an abstract setting is given. Then the Lebesgue multi-dimensional delta measure on time scales is introduced and the Lebesgue delta measure of any single-point set is calculated. When we have a measure, integration theory is available according to the well-known general scheme of the Lebesgue integration process. Finally, we compare the Lebesgue multiple delta integral with the Riemann multiple delta integral. We indicate also a way to define, along with the Lebesgue multi-dimensional delta measure, the nabla and mixed types Lebesgue multi-dimensional measures on time scales.