关于Riemann可积的一个注记

鉴于R iem ann积分对区间的分割和取点的双重任意性要求给应用带来的困难,本文中我们将选择“等距分割”这种特殊的分割方式来定义函数的可积性,即所谓“等分可积”.从表面上看,等分可积性要弱于R iem ann可积性,但本文将证明这两种可积性是等价的.文中还给出了等分可积准则.

A Remark on Riemann Integrability

Riemann integrability requires to consider all possible partitions of the interval and all possible choices of points in each subinterval,which leads difficulties in application. In this paper we define the so-called equi-partition integrability of functions by only selecting a class of special partitions,the equi-partitions of the interval.From the first look,the equi-partition integrability is weaker than Riemann integrability, but it is proved in this paper that these two kinds of integrability are equivalent.A criterion of the equi-partition integrability is also given.