sigma-finite measure

σ有限测度

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

Translation evidence: Literature usage

Meaning in context

A measure mu on X is sigma-finite if X is the union of countably many measurable sets each having finite measure.

若整个测度空间能由可数多个测度有限的可测集覆盖，则该测度是 σ 有限测度。

Example

Lebesgue measure on R\mathbb R is sigma-finite because R=⋃n≥1[−n,n]\mathbb R=\bigcup_{n\ge1}[-n,n], but its total mass is infinite.

Usage and distinctions

Finite measure implies sigma-finiteness; counting measure on an uncountable set is not sigma-finite.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Measure Theory§1.6, paragraph after Definition 1.9 · 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，有限测度与σ有限测度 · Checked 2026-10-14
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