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 (, uniqueness of measuresMathworldPlanetmath extended from a Ļ€-system ( shows that any measure defined on the Ļƒ-algebraMathworldPlanetmath Ļƒā¢(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 pā¢((a,b))=b-a for a<b. The Lebesgue outer measure Ī¼*:š’«ā¢(X)ā†’ā„+āˆŖ{āˆž} is defined as

Ī¼*ā¢(A)=infā”{āˆ‘i=1āˆžpā¢(Ai):Aiāˆˆš’ž,AāŠ†ā‹ƒi=1āˆžAi}. (1)

This is indeed an outer measureMathworldPlanetmath ( (see construction of outer measures) and, furthermore, for any interval of the form (a,b) it agrees with the standard definition of length, Ī¼*ā¢((a,b))=pā¢((a,b))=b-a (see proof that the outer (Lebesgue) measure of an interval is its length (

We show that intervals (-āˆž,a) are Ī¼*-measurable ( Choosing any Ļµ>0 and interval Aāˆˆš’ž the definition of p gives


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


So, from equation (1)

Ī¼*ā¢(E)ā‰„Ī¼*ā¢(Eāˆ©(-āˆž,a))+Ī¼*ā¢(Eāˆ©(a,āˆž)). (2)

Also, choosing any Ļµ>0 and using the subadditivity of Ī¼*,


As Ļµ>0 is arbitrary, Ī¼*ā¢(Eāˆ©(a,āˆž))ā‰„Ī¼*ā¢(Eāˆ©[a,āˆž)) and substituting into (2) shows that


Consequently, intervals of the form (-āˆž,a) are Ī¼*-measurable. As such intervals generate the Borel Ļƒ-algebra and, by Caratheodoryā€™s lemma, the Ī¼*-measurable setsMathworldPlanetmath form a Ļƒ-algebra on which Ī¼* is a measure, it follows that the restrictionPlanetmathPlanetmathPlanetmath of Ī¼* to the Borel Ļƒ-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