complete measure space

完备测度空间

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A measure space is complete if every subset of each measurable null set is measurable.

测度空间若每个可测零测集的任意子集也可测，则称为完备测度空间。

Example

Lebesgue measure on its full Lebesgue sigma-algebra is complete; restriction to the Borel sigma-algebra is not.

Usage and distinctions

Measure completeness concerns null subsets; metric completeness concerns Cauchy limits and is a different property.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Measure Theory§1.7, Definition 1.14 and Theorem 1.15 · Checked 2026-10-14

Translation sources

  1. 李东风（北京大学）— 金融时间序列分析备课笔记，附录EE.3 勒贝格测度，零测集子集与完备性 · Checked 2026-10-14
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