分数维数概念的产生  

The Generation of The Concept of Fractional Dimension

作　　者：江南[1,2] 曲安京[1] JIANG Nan;QU An-jing(Institute for advanced studies in the history of Science,Northwest University,Xi’an 710127,China;College of Science,Xi’an Shiyou University,Xi’an 710065,China)

机构地区：[1]西北大学科学史高等研究院,西安710127 [2]西安石油大学理学院,西安710065

出　　处：《科学技术哲学研究》2020年第4期87-93,共7页Studies in Philosophy of Science and Technology

基　　金：国家社会科学基金重大项目(15ZBD029);国家自然科学基金数学天元基金项目(11926503);陕西省自然科学基金青年项目(2019JQ-869)。

摘　　要：分数维数概念的产生是数学发展历程中的一件重大事件,它在分形理论的建立过程中起着关键性作用,研究它的产生过程对于完善数学的发展历史具有非常重要的意义。为了测量康托尔集的大小,康托尔和雷蒙德率先提出了容度理论,皮亚诺和若尔当相继改进了容度理论,波莱尔和勒贝格则将容度理论升华为测度理论,从而攻克了一些病态函数在黎曼意义下难以求积的难题。受此启迪,卡拉泰奥多里对测度理论进行了公理化探究,在外测度和集函数之间建造了连通彼此的桥梁,还基于q维空间给出了线性测度和p维测度的概念。豪斯多夫则顺势将维数的取值范围推广到非整数,创立了豪斯多夫测度和豪斯多夫维数,从而解决了康托尔集的测量问题。贝西科维奇在豪斯多夫维数的基础上,完善了分数维数的概念。The generation of the concept of fractional dimension is an important event in the development of mathematics.It plays a key role in the establishment of fractal theory.It is of great significance to study its generation process for improving and perfecting the development history of mathematics.In order to measure the size of Cantor set,Cantor and Reymond first put forward the theory of tolerance.Peano and Jordan improved the theory of tolerance one after another,while Borel and Lebesgue sublimed the theory of tolerance into the theory of measurement,thus conquering the difficult problem that ill conditioned functions can’t be integrated in the sense of Riemann.Inspired by this,Carathéodory did axiomatic research on measure theory,built a bridge between outer measure and set function,and gave the concepts of linear measure and p dimension measure based on q dimension space.Hausdorff extended the dimension range to noninteger,and established Hausdorff measure and Hausdorff dimension,which solved the measurement problem of Cantor set.Besicovitch consummated the concept of fractional dimension on the basis of Hausdorff dimension.

关 键 词：康托尔集 容度 测度 分数维数 

分 类 号：N09[自然科学总论—科学技术哲学]