实分析中某些病态函数的探讨

The Discussion of Singular Function in Real Analysis

在实分析的发展过程中,有些反例是相当重要的,因为正是这些重要的反例推动了实分析的发展。文章讨论了在学习过程中发现的一些重要的反例,并对这些反例所起的作用进行了相应的讨论。着重讨论了可测函数的可测复合函数不一定可测的例子,有理点计数测度以及紧空间上的凝聚算子。在最后,讨论了关于收敛定理的两个例子。

In this paper, we discuss some important singular function in real analysis. These functions impulse the development of real analysis. We mainly discuss the example which explains that a measurable function compound a measurable function may be a unmeasurable function. We also discuss the rational counting measure and a coherence operator in the compact space. In the last of this paper, we discuss some examples of convergence theorem in real analysis.