黎曼积分和勒贝格积分区别和联系毕业论文.doc

毕 业 论 文 黎曼积分与勒贝格积分的区别与联系 王建红 答辩委员会主席： 评 阅 人： 2007年06月 Dissertation Submitted to Huangshan University for The Bachelor Degree of Mathematics and Applied Mathematics THE DIFFERENCE AND RALATION BETWEEN RIEMANN CALCULUS AND LEBEGUE CALCULUS By Wang JianHong Supervisor: Vice-Prof． Xiang MingYin May 2007 勒贝格积分与黎曼积分的区别与联系 黄山学院毕业论文 PAGE \* roman ii PAGE \* roman iii 勒贝格积分与黎曼积分的区别与联系 摘 要 本文从微积分的发展过程出发引出了我们已知的黎曼积分，尽管黎曼积分的理论比较完备，但在考虑某些问题时，我们看到了黎曼积分的局限性，并通过具体的例子给予了说明．于是就有了改造黎曼积分的必要性，从而提出了勒贝格积分．本文的中心任务就是从我们已学过的黎曼积分和勒贝格积分的知识来探讨和归纳出两者之间的区别与联系，通过具体比较两者的定义，存在的条件，黎曼积分和勒贝格积分的性质、黎曼可积函数类和勒贝格可积函数类、以及与黎曼积分和勒贝格积分相关的一些定理，并进一步用具体的例子来说明勒贝格积分使一些黎曼积分难以解决的问题变得迎刃而解，最后总结两者之间的区别与联系．并顺便指出，勒贝格积分是黎曼积分的推广，但非黎曼反常积分的推广． 关键词：黎曼积分，勒贝格积分，区别，联系 THE DIFFERENCE AND RALATION BETWEEN RIEMANN CALCULUS AND LEBEGUE CALCULUS ABSTRACT This article begins from the fluxionary calculus developing process which draws out our have known Riemann integral calculus.Although the Riemann integral calculus theory is quite complete， when considered some questions， we saw the Riemann integral calculus very limit． To explain this question through the concrete examples．Therefore it is necessary to point out the Lebesgue integral calculus ．which have the very superiority compared to the Riemann integral calculus． This article’s central task is to discuss and induce the difference and the relation between the Riemann integral calculus and Lebesgue integral calculus from what we have studied the knowledge about the two kinds calculus． Specifically comparing their definition， the existence of conditions， the nature of the Riemann calculus and Lebesgue calculus ，the Riemann integral calculus function class and Lebesgue integral calculus function class， as well as about the Riemann integral calculus and Lebesgue integral calculus there are some theorems． Furthermore through using the conc