几种推广黎曼积分的比较研究

Comparing Several Generalized Riemann Integral

通过证明和反例讨论黎曼积分、直接黎曼积分、黎曼-斯蒂尔切斯积分三者间的联系与区别.结果显示:若函数直接黎曼可积,则它黎曼可积,并且两者积分值相同,但反之不成立;若函数黎曼可积,则任意连续函数关于该函数不一定黎曼-斯蒂尔切斯可积.从讨论结果中还获得直接黎曼可积和黎曼可积各自的一个充分条件.

The relations and distinctions among Riemann integral,directly Riemann integral and RiemannStieltjes integral are discussed with proofs and counterevidence.The discussion shows： if a function is directly Riemann integrabl,then the function is Riemann integrable and the integral values remain unchanged,but the contrary does not hold;if a function is Riemann integrable,then any continuous function is not necessarily Riemann-Stieltjes integrable with respect to the function.Also in this paper,a sufficient condition is obtained for both the directly Riemann integrable and the Riemann integrable.