勒贝格积分创建的思想方法及意义

The Significance and Thinking Way of the Lebesgue Integral

本文认为,Lebesgue积分创建的思路和方法是,以克服Riemann积分在可积性,级数收敛等方面的困难问题为开端,以点集、测度,可测函数等概念的构造为环节和工具,以分割Y轴的逆向思维为突破口而创造性地完成的。Lcbesgue积分创建过程中始终渗透着哲理思维的影响,Lebesgue积分与Riemann积分之间存在着一种和谐、互补的关系。它在控制论、统计力学等数学分支和自然科学部门中有着深刻的应用。

The author thinks that the thinking way established in Lebesgue's integral is a progress of overcoming the limitation of Riemann integral, with the heip of point set,measurable function,and with an unversed thinking of cutting the Y-axis as a break.The Lebesgue integral and the Riemann integral are harmonious and complementary.It has wide applications.