数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (5): 1230-1236.doi: 10.1016/S0252-9602(17)30070-X

• 论文 • 上一篇    下一篇

GENERALIZED RIEMANN INTEGRAL ON FRACTAL SETS

苏峰   

  1. Department of Basic Courses, Guangzhou Maritime University, Guangzhou 510330, China
  • 收稿日期:2016-08-20 修回日期:2017-01-13 出版日期:2017-10-25 发布日期:2017-10-25
  • 作者简介:Feng SU,E-mail:sufeng@gzhmt.edu.cn

GENERALIZED RIEMANN INTEGRAL ON FRACTAL SETS

Feng SU   

  1. Department of Basic Courses, Guangzhou Maritime University, Guangzhou 510330, China
  • Received:2016-08-20 Revised:2017-01-13 Online:2017-10-25 Published:2017-10-25

摘要:

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.

关键词: fractal set, integral, Hausdorff measure, s-set

Abstract:

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.

Key words: fractal set, integral, Hausdorff measure, s-set