Lebesgue measurable set

勒贝格可测集

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

Translation evidence: Literature usage

Meaning in context

A subset E of R^n is Lebesgue measurable if m*(A)=m*(A intersect E)+m*(A minus E) for every subset A of R^n, where m* is Lebesgue outer measure.

若任意测试集合 A 的外测度都等于它在 E 内外两部分的外测度之和，则 E 为勒贝格可测集。

Example

$[0,1]$ and every subset of a Lebesgue null set are Lebesgue measurable.

Usage and distinctions

Lebesgue measurable does not mean Borel: completing Borel measure adds subsets of null sets.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Measure Theory§2.3, Definition 2.10, printed p.17 · 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，勒贝格可测条件 · Checked 2026-10-14

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