勒贝格积分三种定义的等价证明

The Proof Equivalence of the Three Definitions of Lebesgue's Integral

实变函数教材中给出了三种勒贝格积分的定义,这三种定义在形式上存在着很大的差别。为了使初学者能很好的掌握勒贝格积分,这里将讨论这三种勒贝格积分定义的特征,并给出它们之间的等价性证明。从而使初学者能深刻地体会勒贝格积分与黎曼积分的区别,并在创新思维的培养方面受到启发。

Generally,there are three definitions of Lebesgue's integral,but much difference exists in the form of the definitions,which will make troubles for the beginners.In this paper,we will present the equivalence proof of the three definitions of Lebesgue's integral,and discuss the characteristics of the three definitions.From this,we can more profoundly read the difference between the Lebesgue's integral and Riemman's integral,and get enlightenment in creative thinking.