Lebesgue measure

Let $S\subseteq ℝ$, and let $S^{\prime }$ be the complement of $S$ with respect to $ℝ$. The set $S$ is said to be Lebesgue measurable if, for any $A\subseteq ℝ$,

$$m^{*}(A)=m^{*}(A\cap S)+m^{*}(A\cap S^{\prime })$$

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 sets as well all the sets obtained from them by taking countable unions and intersections. However, with aid of the axiom of choice it is possible to construct non-measurable sets.

The Lebesgue measure on ${ℝ}^n$ is the completion (http://planetmath.org/CompletionOfAMeasureSpace) of the $n$-fold product measure of the Lebesgue measure on $ℝ$.

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