关于R可积函数空间的完备化

On the Completion of the Riemann Product Function Space

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:R[a,b]作为L[a,b]的子空间是不完备的,而L[a,b]是完备的,并证明了R[a,b]在L[a,b]中稠密,最后得到了L[a,b]是R[a,b]的完备化空间.

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].