简述从黎曼积分到勒贝格积分的演变

简述从黎曼积分到勒贝格积分的演变

摘要

本文介绍了黎曼积分的概念和可积的充要条件,由于利用定积分的定义来计算很困难从而引入了微积分的基本定理（牛顿-莱布尼茨公式）并展开对微积分的讨论,当注意到黎曼积分与极限可交换的条件太严、不完全是微分运算的逆运算时提出了勒贝格积分并对勒贝格积分的概念、定理、性质和应用及其与黎曼积分相比较所体现出的优越性做出了阐述．
关键字: 微积分；极限；黎曼可积；测度；勒贝格积分

The evolution from Riemann integral to Lebesgue integral
  
 

  ABSTRACT

In this paper, the concept of the Riemann integral and the sufficient and necessary conditions of Riemann integrability are introduced. As using the definition of fixed integral to computing itself is very difficult, so we has brought the basic calculus theorem (Newton-Leibniz formula) and launched the discussion about calculus. When the exchangeable conditions of integral and limit operation are found too strict, and  Riemann integral is not completely differential operator’s inverse operation, we has noticed the Lebesgue integral , including Lebesgue integral concept, theorem, the nature and application . The superiority of Lebesgue integral compared to Riemann integral reflects is also given.
Keywords: Calculus; Limit; Riemann integrability; Measure; Lebesgue integral