滤子定义下的黎曼积分

在一般教材的黎曼积分定义中，黎曼和不是定义在实数或复数域上的，并且黎曼和的极限（即黎曼积分）是在积分区间无限细分情形下的极限，因此这种特殊的极限与数列的极限和函数的极限有着本质上的区别．在定义中对极限的实质阐述不够充分，使学生不容易理解和掌握．我们借鉴国外经典的数学分析教材中的滤子的概念来定义黎曼积分，使定义自然、合理，并和数列极限、函数极限等极限定义有统一的形式．

Definition of Riemann Integral Using Filter

In most textbooks, Riemann sum is not defined on real or complexdomain, and the limit of Riemann sum,Riemann integral, is defined as the limit value of the summation when the integral interval is refined infinitely. It turns out there are essential difference between this limit and limits of series and functions. In addition, this definition does not elaborate the nature of limit clearly, which makes it difficult for students to understand. Therefore, we make use of the concept of filter, which can be found in some classical foreign textbooks in real analysis. Using filter to define Riemann integral is natural and proper. More importantly, its form is unified with limits of series and functions.