Riemann积分与Lebesgue积分

Riemann积分与Lebesgue积分
 
摘  要：本文比较了Riemann积分和Lebesgue积分的定义、基本性质；给出了Lebesgue积分不同建立方式的等级性证明.我们讨论了有限区间和无穷区间上Riemann可积和Lebesgue可积的关系；最后我们将 中有限区间上Riemann可积和Lebesgue可积的结果推广到了高维的欧氏空间中的有界区域.
关键词：R积分；L积分；R可积；L可积；广义R积分.

Riemann integral and Lebesgue integral
 

Abstract：In this paper , we discuss Riemann integral and Lebesgue integral mainly . we compare some  foundamental properties and definitions of these two integrals ; we give a proof of equivalence of three different  definitions of  Lebesgue  integral .  Also, we discuss the  relation between “Riemann  integrable” and “Lebesgue  integrable” on finite and infinite intervals . Finally , we  give  a  proof  showing  that the famous result which says that Riemann  interable can lead  to the Lebesgue integrable in   also holds in finite domains in multi-dimensional space  （ ）.
Key words：Riemann integral;  Lebesgue integral;  Riemann integrable;  Lebesgue interable; generalized Riemann integral.