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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: MULTIPLE LEBESGUE INTEGRATION ON TIME SCALES | MULTIPLE LEBESGUE INTEGRATION ON TIME SCALES MARTIN BOHNERAND GUSEIN SH. GUSEINOV Received 26 January 2006 Revised 17 April 2006 Accepted 18 April 2006 We study the process of multiple Lebesgue integration on time scales. The relationship of the Riemann and the Lebesgue multiple integrals is investigated. Copyright 2006 M. Bohner and G. Sh. Guseinov. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction 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. The main part of this paper is Section 3. There a brief description of the Caratheodory 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. .
