Editing Length (section)

=== Measure theory ===
{{main|Lebesgue measure}}
In measure theory, length is most often generalized to general sets of <math>\mathbb{R}^n</math> via the [[Lebesgue measure]]. In the one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely, the length of an [[Open Interval|open interval]] is first defined as
: <math>\ell(\{x\in\mathbb R\mid a<x<b\})=b-a.</math>
so that the Lebesgue outer measure <math>\mu^*(E)</math> of a general set <math>E</math> may then be defined as<ref>{{cite web|url=http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf|title=Lebesgue Measure|last=Le|first=Dung|url-status=live|archive-url=https://web.archive.org/web/20101130171814/http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf|archive-date=2010-11-30}}</ref>
: <math>\mu^*(E)=\inf\left\{\sum_k \ell(I_k):I_k\text{ is a sequence of open intervals such that }E\subseteq\bigcup_k I_k\right\}.</math>