勒贝格的测度理论成因探析  

A Research on the Factors behind the Establishment of Lebesgue’s Measure Theory

作　　者：李亚亚 LI Yaya(School of Statistics,Xi'an University of Finance and Economics,Xi'an 710100,China)

机构地区：[1]西安财经大学统计学院,西安710100

出　　处：《中国科技史杂志》2022年第3期374-381,共8页The Chinese Journal for the History of Science and Technology

基　　金：国家自然科学基金资助项目“同伦论的历史研究”(项目编号:11501444);陕西省自然科学青年基金资助项目“希尔伯特空间理论的历史研究”(项目编号:2021JQ-763);陕西省教育厅科学基金资助项目“紧算子理论的历史研究”(项目编号:19JK0317)。

摘　　要：测度理论是实变函数的重要内容。勒贝格为改进黎曼积分,从几何观点出发,以博雷尔的测度思想为指导,运用内填外包方法定义可测集,进而建立了他的测度理论。他以此为基础建立了勒贝格积分,这是现代分析学的开端。从外因和内因两个方面,分析研究勒贝格成功创立其测度理论的原因,不仅有助于我们更好地理解勒贝格的数学思想,而且有助于更清楚地呈现实变函数的发展历程。Measure theory is the important content of functions of real variables.In order to improve Riemann integral,Lebesgue established his theory of measure from a geometric point of view.Under the guidance of Borel’s thought of measure,he defined measurable sets by internal and external approximation.Based on his theory of measure,Lebesgue established Lebesgue integral,which is the beginning of modern analysis.From the two aspects of external and internal factors,historical exploring of the reasons of the formation of Lebesgue’s measure theory can not only help us better understand the mathematical thought of Lebesgue,but can also help us present more clearly the evolution of real variable function.

关 键 词：近现代数学史 勒贝格 测度 黎曼积分 勒贝格积分 

分 类 号：N09[自然科学总论—科学技术哲学] O174.12[理学—数学]