Lebesgue measure

Let S, and let S be the complementMathworldPlanetmath of S with respect to . The set S is said to be Lebesgue measurable if, for any A,


where m*(S) is the Lebesgue outer measure of S. If S is Lebesgue measurable, then we define the Lebesgue measure of S to be m(S)=m*(S). The Lebesgue measurable sets include open sets, closed setsPlanetmathPlanetmath as well all the sets obtained from them by taking countableMathworldPlanetmath unions and intersectionsMathworldPlanetmath. However, with aid of the axiom of choiceMathworldPlanetmath it is possible to construct non-measurable sets.

The Lebesgue measure on n is the completionPlanetmathPlanetmath (http://planetmath.org/CompletionOfAMeasureSpace) of the n-fold product measureMathworldPlanetmath of the Lebesgue measure on .

The Lebesgue measure is a formalization of the intuitive notion of length of a set in , an area of a set in 2 and volume in 3, etc. It obeys many properties one would expect from these intuitive notions, such as invariance under translationPlanetmathPlanetmath and rotation.

The Lebesgue measure was introduced by Henri Lebesgue in the first decade of the twentieth century. It became the prototypical example of what later became known simply as measureMathworldPlanetmath, a concept which unified such diverse objects as area, probability, and function.

Title Lebesgue measure
Canonical name LebesgueMeasure
Date of creation 2013-03-22 11:48:24
Last modified on 2013-03-22 11:48:24
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 16
Author bbukh (348)
Entry type Definition
Classification msc 28A12
Classification msc 26A42
Classification msc 03B52
Classification msc 03B50
Related topic Measure
Related topic LebesgueOuterMeasure
Related topic Integral2
Related topic MikowskiInequality
Related topic VitalisTheorem
Related topic BorelSigmaAlgebra
Related topic HausdorffMeasure
Defines Lebesgue measurable