# 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].

#### References

[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) |

#### Comments

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.

#### References

[a1] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) |

**How to Cite This Entry:**

Lebesgue measure.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_measure&oldid=15438