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Riemann積分與Lebesgue積分
Riemann積分與Lebesgue積分
摘 要：本文比較了Riemann積分和Lebesgue積分的定義、基本性質(zhì)；給出了Lebesgue積分不同建立方式的等級(jí)性證明.我們討論了有限區(qū)間和無(wú)窮區(qū)間上Riemann可積和Lebesgue可積的關(guān)系；最后我們將 中有限區(qū)間上Riemann可積和Lebesgue可積的結(jié)果推廣到了高維的歐氏空間中的有界區(qū)域.
關(guān)鍵詞：R積分；L積分；R可積；L可積；廣義R積分.
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.
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