Fatou's lemma

Fatou引理

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

Translation evidence: Literature usage

Meaning in context

For a sequence of nonnegative measurable functions on a measure space, the integral of the pointwise lower limit is at most the lower limit of their integrals.

测度空间上的非负可测函数列，其逐点下极限的积分不大于各积分的下极限，二者可以为无穷。

Example

For fn=n21(0,1/n2)f_n=n^2\mathbf1_{(0,1/n^2)}, ∫lim inf⁡fn=0<1=lim inf⁡∫fn\int\liminf f_n=0<1=\liminf\int f_n.

Usage and distinctions

The inequality may be strict and requires nonnegativity, or an appropriate common integrable lower bound after reduction.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Measure Theory§4.5, Theorem 4.22 and equation (4.7) · 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，Fatou引理 · Checked 2026-10-14
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