关于R可积函数空间的完备化

完备性是度量空间中的一个重要性质,本文运用了实变函数中点集分析的方法及其相关定义和定理讨论了Riemann积分与Lebesgue积分的本质区别在于:Ra,b]作为La,b]的子空间是不完备的,而La,b]是完备的,并证明了Ra,b]在La,b]中稠密,最后得到了La,b]是Ra,b]的完备化空间.

On the Completion of the Riemann Product Function Space

Completeness is an important property in the metric space.In this paper,the essential difference between R a, b] integration and L a, b] integration is discussed with the method of point set-analysis and related defines and theories in Real Function.The difference is that R a, b] as the subspace of L a, b] is incomplete and L a, b] is complete.It also proves that R a, b] is dense in L a, b].Finally,it obtains that L a, b] is the completion space of R a, b].