波莱尔零测集理论思想研究

The Research on Emile Borel's Work Relative to Theory of Zero Measure Set

以法国数学家波莱尔的有关工作为核心,重点考察了他提出零测集的思想背景,提出零测集渐近测度和稀疏度的思想演变过程、思想方法及影响。指出波莱尔在前人工作的基础上于1894年研究函数单演理论时已经有了零测集的思想,在1898年正式提出了零测集的概念,并在此基础上给出了集合测度的概念。正是在函数单演理论研究中,他认识到了零测集的重要性,开始了零测集的分类和性质研究,并在1919年提出了零测集的渐近测度的概念。波莱尔在1935年对渐近测度的概念做适当修改,提出了零测集稀疏度的概念。他为了研究零测集向量和的问题,在1948年给出了3种重要的稀疏度的定义和计算方法。这一工作对弗雷歇、马沙哈尔等人有一定影响。

Theory of zero measure set is an important and difficult question in measure theory This paper is based on Borel＇s work relative to zero measure set. It carefully investigates thought background of proposing zero measure set, thought process of proposing asymptotic measure and rarefaction of proposing zero measure set, and influences on M. Frechet and Jr. Marshall Hall. Based on the investigation, it points out that Borel studied the classification and properties of zero measure set during the research on monogenic function. He gave the concept of asymptotic measure in 1919 and rarefaction in 1935. In 1948, Borel gave three important methods of calculation of rarefaction in order to study the vector sum of zero measure set.