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.