almost everywhere

几乎处处

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

Translation evidence: Literature usage

Meaning in context

A property holds mu-almost everywhere on X when there is a measurable null set N such that the property holds at every point of X minus N.

若存在一个可测零测集，使某性质在该集之外每一点都成立，则称该性质相对于指定测度几乎处处成立。

Example

The rational indicator on $[0,1]$ equals zero Lebesgue almost everywhere, but not at every point.

Usage and distinctions

The exceptional set may be dense and infinite; almost everywhere does not mean everywhere except finitely many points.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Measure Theory§1.7, almost-everywhere paragraph; §3.5 · 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，几乎处处成立段落 · Checked 2026-10-14

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