measurable real function

可测函数

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

Translation evidence: Literature usage

Meaning in context

A real function f on a measurable space (X,A) is measurable when the inverse image of every Borel subset of R belongs to A; equivalently each set {x:f(x)>t} is measurable.

可测空间上的实函数，若每个实数阈值 t 对应的上水平集 {x:f(x)>t} 都可测，则称为可测函数，等价于所有 Borel 集的原像可测。

Example

$\mathbf1_{[0,1]}$ on $\mathbb R$ is measurable despite its endpoint discontinuities.

Usage and distinctions

Measurability depends on specified sigma-algebras and does not require continuity or even a measure value.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Measure Theory§§3.1–3.2, Definitions 3.1, 3.3 · 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.4 勒贝格可测函数，可测函数定义 · Checked 2026-10-14

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