勒贝格积分可加性的严格证明

A Strict Proof of Additive Property for the Lebesuge Integral

文[1]在建立勒贝格积分理论体系时,勒贝格积分可加性是作为从定义出发直接导出的最基本结果,而两个变量和的上确界不小于各自上确界的和这一结果成为完成证明的关键.本文指出了该教材在证明过程中运用的上述上确界大小关系存在反例,并提出了勒贝格积分可加性的新的证明方法,避开了上述上确界大小关系.首先从定义出发证明了非负可测函数勒贝格积分加法的一个次线性性,从而给出了勒贝格积分可加性的严格证明.

When the Lebesgue integral theory was established in article[1],the additive property of the Lebesgue integral was the most basic result derived from the definition directly,and it becomes the key to the proof of completion that the supremum of sum of two variables are not less than the sum of their supremum.In this paper,a counterexample is pointed out against the size relationship of the supremum in this textbook in the course of the proof,and propose a new method to prove the additivity of the Lebesgue integral,which avoid the size relationship of the supremum.In the first,a sublinear property of the Lebesgue integral addition for nonnegative measurable functions is proved,and then a strict proof of additivity of the Lebesgue integral is given.