Lebesgue outer measure


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


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:


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.

Title Lebesgue outer measure
Canonical name LebesgueOuterMeasure
Date of creation 2013-03-22 11:48:15
Last modified on 2013-03-22 11:48:15
Owner yark (2760)
Last modified by yark (2760)
Numerical id 14
Author yark (2760)
Entry type Definition
Classification msc 28A12
Synonym outer measure
Related topic InfimumMathworldPlanetmath
Related topic LebesgueMeasure
Related topic ProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength
Related topic CaratheodorysLemma
Related topic ConstructionOfOuterMeasures