proof of existence of the Lebesgue measure


First, let š’ž be the collectionMathworldPlanetmath of bounded open intervals of the real numbers. As this is a Ļ€-system (http://planetmath.org/PiSystem), uniqueness of measuresMathworldPlanetmath extended from a Ļ€-system (http://planetmath.org/UniquenessOfMeasuresExtendedFromAPiSystem) shows that any measure defined on the σ-algebraMathworldPlanetmath σ⁢(š’ž) is uniquely determined by its values restricted to š’ž. It remains to prove the existence of such a measure.

Define the length of an interval as p⁢((a,b))=b-a for a<b. The Lebesgue outer measure μ*:š’«ā¢(X)ā†’ā„+∪{āˆž} is defined as

μ*⁢(A)=inf⁔{āˆ‘i=1āˆžp⁢(Ai):Aiāˆˆš’ž,AāŠ†ā‹ƒi=1āˆžAi}. (1)

This is indeed an outer measureMathworldPlanetmath (http://planetmath.org/OuterMeasure2) (see construction of outer measures) and, furthermore, for any interval of the form (a,b) it agrees with the standard definition of length, μ*⁢((a,b))=p⁢((a,b))=b-a (see proof that the outer (Lebesgue) measure of an interval is its length (http://planetmath.org/ProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength)).

We show that intervals (-āˆž,a) are μ*-measurable (http://planetmath.org/CaratheodorysLemma). Choosing any ϵ>0 and interval Aāˆˆš’ž the definition of p gives

p⁢(A)=p⁢(A∩(-āˆž,a))+p⁢(A∩(a,āˆž)).

So, choosing an arbitrary set EāŠ†ā„ and a sequence Aiāˆˆš’ž covering E,

āˆ‘i=1āˆžp⁢(Ai)=āˆ‘i=1āˆžp⁢(Ai∩(-āˆž,a))+āˆ‘i=1āˆžp⁢(Ai∩(a,āˆž))≄μ*⁢(E∩(-āˆž,a))+μ*⁢(E∩(a,āˆž)).

So, from equation (1)

μ*⁢(E)≄μ*⁢(E∩(-āˆž,a))+μ*⁢(E∩(a,āˆž)). (2)

Also, choosing any ϵ>0 and using the subadditivity of μ*,

μ*⁢(E∩(a,āˆž))≄μ*⁢(E∩(a-ϵ,āˆž))-μ*⁢(E∩(a-ϵ,a+ϵ))≄μ*⁢(E∩[a,āˆž))-μ*⁢((a-ϵ,a+ϵ))=μ*⁢(E∩[a,āˆž))-2⁢ϵ.

As ϵ>0 is arbitrary, μ*⁢(E∩(a,āˆž))≄μ*⁢(E∩[a,āˆž)) and substituting into (2) shows that

μ*⁢(E)≄μ*⁢(E∩(-āˆž,a))+μ*⁢(E∩[a,āˆž)).

Consequently, intervals of the form (-āˆž,a) are μ*-measurable. As such intervals generate the Borel σ-algebra and, by Caratheodory’s lemma, the μ*-measurable setsMathworldPlanetmath form a σ-algebra on which μ* is a measure, it follows that the restrictionPlanetmathPlanetmathPlanetmath of μ* to the Borel σ-algebra is itself a measure.

Title proof of existence of the Lebesgue measureMathworldPlanetmath
Canonical name ProofOfExistenceOfTheLebesgueMeasure
Date of creation 2013-03-22 18:33:14
Last modified on 2013-03-22 18:33:14
Owner gel (22282)
Last modified by gel (22282)
Numerical id 10
Author gel (22282)
Entry type Proof
Classification msc 26A42
Classification msc 28A12