simple measurable function

简单函数

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

Translation evidence: Literature usage

Meaning in context

A simple function on a measurable space is a real measurable function taking finitely many values, equivalently a finite linear combination of indicators of measurable sets.

可测空间上只取有限多个实值的可测函数称为简单函数，也可写成有限个可测集指示函数的线性组合。

Example

$2\mathbf1_{[0,1]}-3\mathbf1_{(1,2]}$ is simple and has integral $2-3=-1$.

Usage and distinctions

Simple functions may be constant on irregular measurable sets; step functions usually use interval or rectangle pieces.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Measure Theory§3.4, Definition 3.11 · 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，简单函数与阶梯函数 · Checked 2026-10-14

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