In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n-spaces. For lower dimensions ${\displaystyle n=1,2,{\text{or }}3}$, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ${\displaystyle A}$ is here denoted by ${\displaystyle \lambda (A)}$.

Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation Intégrale, Longueur, Aire in 1902.

For any interval ${\displaystyle I=[a,b]}$, or ${\displaystyle I=(a,b)}$, in the set ${\displaystyle \mathbb {R} }$ of real numbers, let ${\displaystyle \ell (I)=b-a}$ denote its length. For any subset ${\displaystyle E\subseteq \mathbb {R} }$, the Lebesgue outer measure ${\displaystyle \lambda ^{\!*\!}(E)}$ is defined as an infimum

$${\displaystyle \lambda ^{\!*\!}(E)=\inf \left\{\sum _{k=1}^{\infty }\ell (I_{k}):{(I_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of open intervals with }}E\subset \bigcup _{k=1}^{\infty }I_{k}\right\}.}$$

The above definition can be generalised to higher dimensions as follows. For any rectangular cuboid ${\displaystyle C}$ which is a Cartesian product ${\displaystyle C=I_{1}\times \cdots \times I_{n}}$ of open intervals, let ${\displaystyle \operatorname {vol} (C)=\ell (I_{1})\times \cdots \times \ell (I_{n})}$ (a real number product) denote its volume. For any subset ${\displaystyle E\subseteq \mathbb {R^{n}} }$,

$${\displaystyle \lambda ^{\!*\!}(E)=\inf \left\{\sum _{k=1}^{\infty }\operatorname {vol} (C_{k}):{(C_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of products of open intervals with }}E\subset \bigcup _{k=1}^{\infty }C_{k}\right\}.}$$

A set ${\displaystyle E}$ satisfies the Carathéodory criterion whenever, for every ${\displaystyle A\subseteq \mathbb {R^{n}} }$, we have:

$${\displaystyle \lambda ^{\!*\!}(A)=\lambda ^{\!*\!}(A\cap E)+\lambda ^{\!*\!}(A\cap E^{\complement }).}$$