Lebesgue integral

勒贝格积分

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

Translation evidence: Literature usage

Meaning in context

For nonnegative measurable f on a measure space, its Lebesgue integral is the supremum of integrals of nonnegative simple functions below f. For signed f, subtract the negative-part integral from the positive-part integral when they are not both infinite.

测度空间上非负可测函数的勒贝格积分，是其下方非负简单函数积分的上确界；一般实值函数按正负部分相减定义，但不允许两个无穷积分相减。

Example

On [0,2][0,2], ∫1Q=0\int\mathbf1_{\mathbb Q}=0 since the rational set has measure zero.

Usage and distinctions

Having a defined extended integral differs from being integrable, which requires a finite integral of the absolute value.

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Definition sources

  1. John K. Hunter — Measure Theory§§4.1–4.3, Definitions 4.1, 4.4, 4.8 · 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.6 勒贝格积分，简单函数积分与一般函数积分 · Checked 2026-10-14
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