construction of outer measures

The following theorem is used in measure theory to construct outer measuresMathworldPlanetmathPlanetmath ( on a set X, starting with a non-negative functionMathworldPlanetmath on a collection of subsets of X. For example, if we take X to be the real numbers, 𝒞 to be the collection of bounded open intervalsDlmfPlanetmath of and define p by p((a,b))=b-a for real numbers a<b, then the Lebesgue outer measure is obtained.


Let X be a set, C be a family of subsets of X containing the empty set and p:CR{} be a function satisfying p()=0. Then the function μ*:P(X)R{} defined by

μ*(A)=inf{i=1p(Ai):Ai𝒞,Ai=1Ai} (1)

is an outer measure.


The definition of μ* immediately gives μ*(A)μ*(B) for sets AB, and if A= then we can take Ai= in (1) to obtain μ*()ip()=0, giving μ*()=0. Only the countable subadditivity of μ* remains to be shown. That is, if Ai is a sequence in 𝒫(X) then

μ*(iAi)iμ*(Ai). (2)

To prove this inequalityMathworldPlanetmath, we may restrict to the case where μ*(Ai)< for each i so that, choosing any ϵ>0, equation (1) says that there exists a sequence Ai,j𝒞 such that AijAi,j and,


As iAii,jAi,j, equation (1) defining μ* gives


As ϵ>0 is arbitrary, this proves subadditivity (2). ∎

Although this result is rather general, placing few restrictions on the function p, there is no guarantee that the outer measure μ* will agree with p for the sets in 𝒞 nor that 𝒞 will consist of μ*-measurable ( sets.

For example, if X=, 𝒞 consists of the bounded open intervals, and p((a,b))=(b-a)2 for real numbers a<b, then μ*((a,b))=0p((a,b)).

Alternatively if p((a,b))=b-a for all a<b then it follows that μ*((a,b))=b-a so


and (0,1) is not μ*-measurable.

Title construction of outer measures
Canonical name ConstructionOfOuterMeasures
Date of creation 2013-03-22 18:33:17
Last modified on 2013-03-22 18:33:17
Owner gel (22282)
Last modified by gel (22282)
Numerical id 10
Author gel (22282)
Entry type Theorem
Classification msc 28A12
Related topic OuterMeasure
Related topic LebesgueOuterMeasure
Related topic CaratheodorysLemma