# Lebesgue measure

in $\mathbf R ^ {n}$

A countably-additive measure $\lambda$ which is an extension of the volume as a function of $n$- dimensional intervals to a wider class ${\mathcal A}$ of sets, namely the Lebesgue-measurable sets. The class ${\mathcal A}$ contains the class ${\mathcal B}$ of Borel sets (cf. Borel set) and consists of all sets of the form $A \cup B$ where $B \subset B _ {1}$, $A , B _ {1} \in {\mathcal B}$ and $\lambda ( B _ {1} ) = 0$. One has for any $A \in {\mathcal A}$,

$$\tag{* } \lambda ( A) = \inf \sum _ { j } \lambda ( I _ {j} ) ,$$

where the infimum is taken over all possible countable families of intervals $\{ I _ {j} \}$ such that $A \subset \cup I _ {j}$. Formula (*) makes sense for every $A \subset \mathbf R ^ {n}$ and defines a set function $\lambda ^ {*}$( which coincides with $\lambda$ on ${\mathcal A}$), called the outer Lebesgue measure. A set $A$ belongs to ${\mathcal A}$ if and only if

$$\lambda ( I) = \lambda ^ {*} ( A \cap I ) + \lambda ^ {*} ( I \setminus A )$$

for every bounded interval $I$; for all $A \subset \mathbf R ^ {n}$,

$$\lambda ^ {*} ( A) = \inf \{ {\lambda ( U ) } : {A \subset U , U \textrm{ is o p e n } } \} ,$$

and for all $A \in {\mathcal A}$,

$$\lambda ( A) = \lambda ^ {*} ( A) = \ \sup \{ {\lambda ( F ) } : {A \supset F , F \textrm{ is compact } } \} ;$$

if $\lambda ^ {*} ( A) < \infty$, then the last equality is sufficient for the membership $A \in {\mathcal A}$; if $O$ is an orthogonal operator in $\mathbf R ^ {n}$ and $a \in \mathbf R ^ {n}$, then $\lambda ( OA + a ) = \lambda ( A)$ for any $A \in {\mathcal A}$. The Lebesgue measure was introduced by H. Lebesgue .

How to Cite This Entry:
Lebesgue measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_measure&oldid=47602
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article