黎曼积分与勒贝格积分的区别与联系摘要：黎曼积分和勒贝格积分分别是数学分析及实变函数的核心内容。本文从学过的黎曼积分和勒贝格积分的知识出发，探讨和归纳出两者之间的区别与联系。从积分的定义，积分的存在条件，积分的性质，可积函数类，积分相关定理五个方面阐述了黎曼积分与勒贝格积分的区别。通过对勒贝格积分与黎曼积分发展过程的分析与探讨归纳出二者的区别与联系。关键词：黎曼积分；勒贝格积分；区别；联系。Abstract：RiemannintegralandtheLebesgueintegralwhichisthecorecontentofmathematicalanalysisandtherealvariablefunction.ThispaperstudiedtheRiemannintegralandtheLebesgueintegralknowledgeof,anddiscussesthedifferenceandconnectionbetweenthetwo.Fromthedefinitionofintegral,thecontinuityofintegrablefunction,theadditivityofintegral,integrallimittheorem,fiveaspectsofNewton-LeibnizformuladescribesthedistinctionsbetweenRiemannintegralandLebesgueintegral.ThroughtheanalysisandDiscussionontheLebesgueintegralandtheRiemannintegraldevelopmentprocessinducesthedifferenceandconnectionbetweenthetwo.Keyword：Riemannintegral；Lebesgueintegral；difference；relationship目录1引言.............................................................................................................................31.1微积分的发展史..............................................................................................31.2黎曼积分与勒贝格积分的引入......................................................................32黎曼积分与勒贝格积分的定义.................................................................................52.1黎曼积分的定义..............................................................................................52.2勒贝格积分的定义..........................................................................................62.3黎曼积分和勒贝格积分定义区别与联系......................................................73黎曼积分与勒贝格积分存在条件.............................................................................73.1黎曼可积的条件..............................................................................................73.2勒贝格可积条件..............................................................................................94黎曼积分与勒贝格积分的性质.................................................................................94.1黎曼积分的性质..............................................................................................94.2勒贝格积分的性质........................................................................................115黎曼可积函数类与勒贝格可积函数类...................................................................125.1黎曼可积函数类............................................................................................125.2勒贝格可积函数类.......................................................................................126黎曼积分与勒贝格积分相关定理...........................................................................136.1与黎曼积分相关的定理...............................................................................136.2与勒贝格积分相关的定理...........................................................................147黎曼积分与勒贝格积分的主要联系.......................................................................158总结和展望................................................................................................................
