monotone convergence theorem

非负单调收敛定理

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

Translation evidence: Literature usage

Meaning in context

If measurable functions 0<=f_n increase pointwise to f on a measure space, then their integrals increase to the integral of f, allowing the value infinity.

测度空间上的非负可测函数列逐点单调递增于 f 时，各函数积分的极限等于 f 的积分，允许为正无穷。

Example

On $[0,1]$, $f_n(x)=\min(nx,1)$ increases to $1$ except at zero, and $\int f_n=1-1/(2n)\to1$.

Usage and distinctions

No single integrable dominating function is required; nonnegativity and monotonicity replace that hypothesis.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Measure Theory§4.2, Theorem 4.6 · 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，非负单调收敛定理（Levi定理） · Checked 2026-10-14

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