Riemann积分与Lebesgue积分之比较  

Comparison Between Riemann Integral and Lebesgue Integral

作　　者：洪云飞[1] 杨超[2] 

机构地区：[1]长江大学期刊社,湖北荆州434020 [2]长江大学信息与数学学院,湖北荆州434020

出　　处：《襄樊学院学报》2004年第2期6-10,共5页Journal of Xiangfan University

摘　　要：研究了Riemann积分与Lebesgue之间的关系,在给出了正常Riemann积分与Lebesgue积分的联系的同时,重点研究了广义Riemann积分与Lebesgue积分的关系,即函数f(x)在[a,b]上Riemann可积时,f(x)在[a,b]上也Lebesgue可积,并且两积分分值相等;但广义Riemann积分与Lebesgue积分之间的关系则不尽然.当无穷积分或瑕积分在区间绝对收敛时,则函数f(x)在此区间也Lebesgue可积,并且两积分分值相等,当无穷积分或瑕积分在区间条件收敛时,则函数f(x)在此区间不Lebesgue可积.This article researches the relations between Riemann integral and lebesgue integral.While giving out the relations between Riemann integral and Lebesgue integral,the paper puts the relations between improper Riemann integral and Lebesgue integral.That in when function )(xf is Riemann integrabel in closed interval ],[ba,it also Lebesgue integrabel in closed interval ],[ba, and the value of the two integral is infinite integral or improper integral of limitless function is absolutely convergent in the interval.function )(xf is Lebesgue integrabel in the interval,and the value of the two integral is equivalent;While infinite integral or improper integral of limitless function is conditionally convergent in the interval.function )(xfis not Lebesgue integrabel in the interval.

关 键 词：RIEMANN积分 LEBESGUE积分 绝对收敛 条件收敛 

分 类 号：O172.2[理学—数学]