浅谈勒贝格积分

摘要

本文以勒贝格积分的形成思想为线索先指出黎曼积分存在缺陷，然后进1步针对黎曼积分缺陷当中的两大不足进行阐述，为了改进积分理论，勒贝格重新建立了1套积分理论，而这套理论的核心关键在于勒贝格引入了测度的概念。本文简略地介绍和解释了测度理论的主要概念，指出由测度引入扩大了可积范围带来了质的飞跃以及勒贝格积分对于黎曼积分的替代性和优越性。

关键字：黎曼积分；测度；勒贝格积分；替代性；优越性

On Lebesgue integral

Abstract

The thread of this paper is showed by pointing out that the flaws of Riemann integral exists firstly, then describing the two major deficiencies among them. In order to improve integration theory, Lebesgue had reestablished a set of integral theory, and the core of this theory lies in introducing the concept of Lebesgue measure. In this paper, the main concepts of the theory on measure are introduced and explained briefly. Besides this, we point out that introducing the measure has led to the expanding of the integrable range , thus it has brought a qualitative leap and showed the replacement and superiority to Lebesgue integral than the Riemann integral .

Keywords: Riemann integral; measure; Lebesgue integra；quality of replacing；superiority

目 录

前言 3

1 RIEMANN积分的不足4

1.1 狄利克雷函数不可积4

1.2 Riemann积分（以下简称R积分）的两个理论缺陷4

1.2.1 R可积函数对连续性的要求4

1.2.2 R积分与极限可交换的条件5

2 测度5

2.1 什么叫测度5

2.2 为什么引入测度的概念5

2.3 测度的计算6

2.3.1 外测度的定义6

2.3.2 外测度的性质6

2.4 如何判断可测集7

2.5 可测集中两个重要的概念 7

2.5.10测度集8

2.5.2几乎处处8

3 可测函数8

3.1 可测函数的定义8

3.2可测函数的判断条件8

3.3 可积函数类的扩大9

4 Lebesgue积分10

5 Lebesgue积分对Riemann积分的替代性10

6 Lebesgue积分和Riemann积分相比的优越性11

6.1 L可积函数可积的要求11

6.2 L积分与极限可交换的条件11

7 结束语13

8 参考文献14

9 致谢15