Riemann integral

黎曼积分

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

Translation evidence: Literature usage

Meaning in context

For a bounded real function on [a,b], the Riemann integral is the common value of the supremum of its lower partition sums and the infimum of its upper partition sums, when those values agree.

闭区间上的有界实函数，若所有分割的下和之上确界与上和之下确界相等，此共同值称为黎曼积分。

Example

For $f(x)=x$ on $[0,1]$, n equal subintervals give lower and upper sums $(n-1)/(2n)$ and $(n+1)/(2n)$, hence integral $1/2$.

Usage and distinctions

On $[0,1]$, the rational indicator is Lebesgue integrable but its Riemann lower and upper integrals are $0$ and $1$.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in One Variable§4.2, equations (4.2.1)–(4.2.7) · 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.6 勒贝格积分，与黎曼积分的关系 · Checked 2026-10-14

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