Riemann积分与Lebesgue积分的比较

The Comparison of Riemann Integral and Lebesgue Integral

从Riemann积分与Lebesgue积分的定义、性质、积分与极限交换次序及微积分基本定理等方面进行比较,并给出Lebesgue积分下的积分中值定理及证明,讨论了Lebesgue积分和Riemann积分二者之间的关系。最后,通过二者在广义积分方面的比较,说明Lebesgue积分在广义积分方面并不是Riemann积分的推广。

This paper compared Riemann integral and Lebesgue integral in the respects of definition,nature,exchanging the order of integral and limit,the basic theorem of Calculus etc.and gave Mean-Value theorem for Lebesgue integral.The relation of Riemann integral and Lebesgue integral is investigated.At last the comparison in generalized integral proves that Lebesgue integral is not the popularization of Riemann integral in generalized integral.