从勒贝革积分理论的建立和发展看函数黎曼可积条件的推广

On the Extending of Integral Conditions of Riemann Function According to the Foundation and Development of Lebesgue Integral

我们知道,连续函数(continuous function)一定可积(integral),进一步研究又知道,有界函数(limitary function)且只有有限多个不连续点(discontinuous point),函数一定可积,那么,函数的可积条件能否进一步推广呢?本文从以测度论(measure theory)为基础的勒贝革积分理论(Lebesgue integral)的建立和发展过程中,探讨了这一问题。

As it is known, continuous functions must be integrabel. And if limitary functions have but only have numerable discontinuous points, they must be integrabel. Then can the integral conditions of functions be further extended？ This question is discussed in this paper according to the foundation and development of Lebesgue Integral which is based on Measure Theory.