PlanetMath: Lebesgue measure

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Lebesgue measure

(Definition)

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

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

where $m^{*}(S)$ is the Lebesgue outer measure of . If is Lebesgue measurable, then we define the Lebesgue measure of 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.

Lebesgue measure on $\mathbb{R}^n$ is the -fold product measure of Lebesgue measure on $\mathbb{R}$ .

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.

"Lebesgue measure" is owned by bbukh. [ full author list (2) | owner history (1) ]

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Also defines:

Lebesgue measurable

Keywords:

real analysis

Pronunciation (guide):

Lebesgue: /l*-beg''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''/

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Cross-references: function, area, objects, measure, Henri Lebesgue, product measure, intersections, unions, countable, closed sets, open sets, Lebesgue outer measure, complement
There are 41 references to this entry.

This is version 8 of Lebesgue measure, born on 2001-10-18, modified 2006-08-03.
Object id is 348, canonical name is LebesgueMeasure.
Accessed 21080 times total.

Classification:

AMS MSC:	26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)
	28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)

Pending Errata and Addenda

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