Lebesgue measure

勒贝格测度

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

Translation evidence: Literature usage

Meaning in context

Lebesgue measure on R^n is the complete measure obtained by restricting the countable rectangular-cover outer measure to the sets satisfying the Carathéodory splitting criterion.

欧氏空间上的勒贝格测度，是把可数矩形覆盖定义的外测度限制在满足卡拉泰奥多里分割条件的集合上所得的完备测度。

Example

The rectangle [1,3]×[0,4][1,3]\times[0,4] has measure (3−1)(4−0)=8(3-1)(4-0)=8.

Usage and distinctions

Outer measure is defined for every subset; Lebesgue measure is defined on the Lebesgue sigma-algebra.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Measure Theory§2.3, Definitions 2.8 and 2.10, Theorem 2.9, printed pp.14–17 · 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，可测集及 m(E)=m*(E) · Checked 2026-10-14
Edit source / propose a PRReport a correction

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.