dominated convergence theorem

控制收敛定理

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

Translation evidence: Literature usage

Meaning in context

If measurable f_n converge almost everywhere to a measurable f and |f_n|<=g almost everywhere for one nonnegative integrable g, then f is integrable and the integrals of f_n converge to that of f.

可测函数列几乎处处收敛于可测函数 f，且绝对值由同一个非负可积函数控制时，f 可积并可交换极限与积分。

Example

On $[0,1]$, $f_n(x)=x^n$ is bounded by $1$ and tends to $0$ almost everywhere; $\int f_n=1/(n+1)\to0$.

Usage and distinctions

Pointwise convergence alone is insufficient: n squared times the indicator of (0,1/n squared) tends to zero but has integral one.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Measure Theory§4.5, Theorem 4.24; §4.6, a.e. version · 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.7 极限与积分的交换，控制收敛定理 · Checked 2026-10-14

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