Riemann積分與Lebesgue積分

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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