有限测度空间导出的度量空间及其完备与可分性

The Metric Space Induced by a Finite Measure Space and its Completeness and Separability

设（Ｘ，Ａ，μ）是一个有限测度空间。本文引入了由（Ｘ，Ａ，μ）导出的度量空间（Ａ，ρ），研究了（Ａ，ρ）的完备性、可分性等若干性质，获得了下列结果：（１）（Ａ，ρ）是完备的度量空间；（２）（Ａ，ρ）可分的充要条件是（Ｘ，Ａ，μ）为μ－可分；（３）Ａ。在Ａ中稠密当且仅当Ａ。在Ａ中μ－稠密。

Let (X, A. μ) be a finite measure space. In this paper, the metric space (A, ρ) induced by (X,A, μ) is introduced and some properties of (A, ρ) are researched. The following three results are obtained.() (A. ρ) is a complete metric space.(2) It is the necessary and sufficient condition under which (A, ρ) is separable that (X. A. μ) is μseparable.(3) A0 is dense in A if and only if A0 is μ-dense in A.