黎曼相关积分研究

Research on Riemann integral and Related

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.

In this paper, relations among Riemann integral,directly Riemann integral and Riemann-Stieltjes integral are discussed. With proofs and counter examples, we show that directly Riemann integrable implies Riemann integrable with same value, but the reverse is not true; a continuous function is not necessarily Riemann-Stieltjes integrable with respect to a Riemann integrable function. We obtain also sufficient conditions for directly Riemann integrable and Riemann integrable.