Lebesgue outer measure

外测度

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

Translation evidence: Literature usage

Meaning in context

The Lebesgue outer measure of any E in R^n is the infimum of sums of rectangular volumes over all countable rectangle families covering E.

欧氏空间任意子集的勒贝格外测度，是所有覆盖该集合的可数矩形族的体积和之下确界。

Example

For a countable set {xk}\{x_k\}, coverings with total volume below any ε>0\varepsilon>0 give outer measure zero.

Usage and distinctions

An outer measure is countably subadditive; countable additivity is guaranteed only after restricting to its measurable sets.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Measure Theory§2.1, Definition 2.2 and Theorem 2.4 · 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，外测度覆盖公式 · Checked 2026-10-14
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