Lebesgue measure

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A countably-additive measure which is an extension of the volume as a function of -dimensional intervals to a wider class of sets, namely the Lebesgue-measurable sets. The class contains the class of Borel sets (cf. Borel set) and consists of all sets of the form where , and . One has for any ,


where the infimum is taken over all possible countable families of intervals such that . Formula (*) makes sense for every and defines a set function (which coincides with on ), called the outer Lebesgue measure. A set belongs to if and only if

for every bounded interval ; for all ,

and for all ,

if , then the last equality is sufficient for the membership ; if is an orthogonal operator in and , then for any . The Lebesgue measure was introduced by H. Lebesgue [1].


[1] H. Lebesgue, "Intégrale, longeur, aire" , Univ. Paris (1902) (Thesis)
[2] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)
[3] P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[4] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)


The Lebesgue measure is a very particular example of a Haar measure, of a product measure (when ) and of a Hausdorff measure. Actually it is historically the first example of such measures.


[a1] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
How to Cite This Entry:
Lebesgue measure. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article