Fubini's theorem

Fubini定理

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

Translation evidence: Literature usage

Meaning in context

For sigma-finite measure spaces X and Y, an integrable real or complex function on their product has integrable sections almost everywhere, and its product integral equals either iterated integral.

两个 σ 有限测度空间上的乘积可积函数，其截面几乎处处可积，而且重积分等于任一顺序的累次积分。

Example

On [0,1]2[0,1]^2, ∫ ⁣∫xy dx dy=(1/2)(1/2)=1/4\int\!\int xy\,dx\,dy=(1/2)(1/2)=1/4.

Usage and distinctions

Fubini uses absolute integrability for signed functions; Tonelli permits a nonnegative integrand with an infinite integral.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Measure Theory§5.6, Theorem 5.18 · 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.8 重积分与累次积分，Fubini定理 · Checked 2026-10-14
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