proof that the outer (Lebesgue) measure of an interval is its length

We begin with the case in which we have a bounded interval, say [a,b]. Since the open intervalPlanetmathPlanetmath (a-ε,b+ε) contains [a,b] for each positive number ε, we have m*[a,b]b-a+2ε. But since this is true for each positive ε, we must have m*[a,b]b-a. Thus we only have to show that m*[a,b]b-a; for this it suffices to show that if {In} is a countableMathworldPlanetmath open cover by intervals of [a,b], then


By the Heine-Borel theorem, any collectionMathworldPlanetmath of open intervals [a,b] contains a finite subcollection that also cover [a,b] and since the sum of the lengths of the finite subcollection is no greater than the sum of the original one, it suffices to prove the inequalityMathworldPlanetmath for finite collections {In} that cover [a,b]. Since a is contained in In, there must be one of the In’s that contains a. Let this be the interval (a1,b1). We then have a1<a<b1. If b1b, then b1[a,b], and since b1(a1,b1), there must be an interval (a2,b2) in the collection {In} such that b1(a2,b2), that is a2<b1<b2. Continuing in this fashion, we obtain a sequence (a1,b1),,(ak,bk) from the collection {In} such that ai<bi-1<bi. Since {In} is a finite collection our process must terminate with some interval (ak,bk). But it terminates only if b(ak,bk), that is if ak<b<bk. Thus

l(In) l(ai,bi)

since ai<bi-1. But bk>b and a1<a and so we have bk-a1>b-a, whence l(In)>b-a. This shows that m*[a,b]=b-a.

If I is any finite interval, then given ε>0, there is a closed intervalJI such that l(J)>l(I)-ε. Hence


where by I¯ we the topological closure of I. Thus for each ε>0, we have l(I)-ε<m*Il(I), and so m*I=l(I).

If now I is an unboundedPlanetmathPlanetmath interval, then given any real number Δ, there is a closed interval JI with l(J)=Δ. Hence m*Im*J=l(J)=Δ. Since m*IΔ for each Δ, it follows m*I==l(I).


Royden, H. L. Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.

Title proof that the outer (Lebesgue) measure of an interval is its length
Canonical name ProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength
Date of creation 2013-03-22 14:47:04
Last modified on 2013-03-22 14:47:04
Owner Simone (5904)
Last modified by Simone (5904)
Numerical id 6
Author Simone (5904)
Entry type Proof
Classification msc 28A12
Related topic LebesgueOuterMeasure