关于可测空间中非负可测函数序列积分的不等式

An Inequality for the Integral of a Nonnegative Measurable Function Sequence in the Measurable Space

Xie在文献[1]给出了一个关于Borel-Cantelli引理的重要的双边不等式,对Borel-Cantelli引理做了一个新的推广.胡舒合、王学军等在文献[2]中指出了在文献[1]中定理的证明及举例有错误,给出了定理的正确证明和例子的详细分析.本文在上述文献的基础之上,主要通过对文献[1]中双边不等式的研究,将Xie的结果推广到可测空间中的非负可测函数序列,得到相应的不等式.

A new extension of the Borel-Cantelli lemma is obtained by Xie[ 1 ] , and an important bilateral inequality on the Borel-Cantelli lemma is given in Xie [ 1 ]. But Hu [ 2 ] pointed out that there were some problems in Xie [ 1 ] , and gave the correct proof of the theory. In this paper, we will expand the Xie＇ s results to the measurable space of nonnegative measurable function sequence by the bilateral inequality in [ 1 ].