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单词 LebesgueOuterMeasure
释义

Lebesgue outer measure


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extension theorem

Let S be a subset of , let L(I) be the traditional definition of the length of an interval I: If I=(a,b), then L(I)=b-a. Finally, let M be the set consisting of the values

ACL(A)

for all possible countableMathworldPlanetmath collectionsMathworldPlanetmath of open intervals C that covers S (that is, SC).Then the Lebesgue outer measure of S is defined by:

m*(S)=inf(M)

Note that (,𝒫(),m*) is an outer measureMathworldPlanetmath space (http://planetmath.org/OuterMeasure2). In particular:

  • Lebesgue outer measure is defined for any subset of (and 𝒫() is a σ-algebra).

  • m*(A)0 for any A, and m*()=0.

  • If A and B are disjoint sets, then m*(AB)m*(A)+m*(B). More generally, if Ai is a countable sequence of disjoint sets, then m*(Ai)m*(Ai). This property is known as countable subadditivity and is weaker than countable additivityMathworldPlanetmath. In fact, m* is not countably additive.

Lebesgue outer measure has other nice properties:

  • The outer measure of an interval is its length: m*((a,b))=b-a.

  • m* is translation-invariant. That is, if we define A+y to be the set {x+y:xA}, we have m*(A)=m*(A+y) for any y.

The outer measure satisfies all the axioms of a measure except (countable) additivity. However, it is countably additive when one restricts to at least the Borel sets, as this is the usual construction of Borel measure. This result is roughly contained in the Caratheodory Extension theorem.

TitleLebesgue outer measure
Canonical nameLebesgueOuterMeasure
Date of creation2013-03-22 11:48:15
Last modified on2013-03-22 11:48:15
Owneryark (2760)
Last modified byyark (2760)
Numerical id14
Authoryark (2760)
Entry typeDefinition
Classificationmsc 28A12
Synonymouter measure
Related topicInfimumMathworldPlanetmath
Related topicLebesgueMeasure
Related topicProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength
Related topicCaratheodorysLemma
Related topicConstructionOfOuterMeasures
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