关于Lebesgue积分两种定义方法的比较

Lebesgue积分有着各种不同的等价定义方法,本文就“划分法”与“逼近法”这两种定义方法进行比较。前者先对可测集进行划分,再类似于R积分的定义方法,利用达布上、下和给出L积分,这样定义便于理解,但不利于L积分中三大核心定理的展开;后者则用简单可测函数来逼近可测函数,虽然这样定义较为抽象,不易理解,但整个过程简洁、明了,且对L积分中三大定理的研究是有利的。

Comparison between Two Definitions of Lebesgue Integral

There are many methods to define Lebesgue integral. Here we compare the methods of "Partition" with "Approximation"Just as Riemann integral, the former defines the Lebesgue integral by partitioning measurable sets and constructing Darboux's sum. It is very natural but unfavorable to bring out the key theories of Lebesgue integral. The latter defines the Lebesgue integral of a general function to be the limit if integrals of a sequence of simple functions that converges to it. The latter is less abstract, but it is advantageous to bring out three key theorems of Lebesgue integral.