实变函数论中勒贝格测度的计算方法

The Calculation Methods of Lebesgue Measure in Real Variable Function

摘要: 众所周知，实变函数论是大学数学分析学中一门非常重要的课程，勒贝格测度是本课程的核心内容。迄今为止，很少有文献介绍如何计算勒贝格可测集的测度。本文针对$R^n$ 空间中的可测集，建立测度的计算方法。首先，基于开集构造定理，讨论了一维和高维开集的计算，并建立了测度的计算公式。其次，通过补集关系，将闭集测度转化为开集测度的计算问题，结合测度的单调增加性质完成对闭集计算方法的建立。最后，对一般高维可测集，本文提供了计算思路和方法。

Abstract: It is well known that the theory of variable real function is a very important course in mathematical analysis in college, and Lebesgue measure is the core content of this course. Up to now, there is little study on how to calculate the Lebesgue measure of measurable sets. In this paper, we focus on the calculation methods of measurable sets. Firstly, based on the construction theorems of open sets, the computation of one-dimensional and higher-dimensional open sets are discussed, and formulas for calculating their measures are derived. Secondly, by leveraging the complement relationship, the problems of computing the measure of closed sets are transformed into calculating the measure of open sets, and the methods for closed sets are established by combining the monotonic increasing property of measures. Finally, for general higher-dimensional measurable sets, this paper provides computational ideas and methods.