outer measure

Definition [1, 2, 3] Let X be a set, and let 𝒫(X) be the power setMathworldPlanetmath of X. An outer measureMathworldPlanetmathPlanetmath on X is a function μ:𝒫(X)[0,] satisfying the properties

  1. 1.


  2. 2.

    If AB are subsets in X, then μ(A)μ(B).

  3. 3.

    If {Ai} is a countableMathworldPlanetmath collectionMathworldPlanetmath of subsets of X, then


Here, we can make two remarks. First, from (1) and (2), it follows that μ is a positive function on 𝒫(X). Second, property (3) also holds for any finite collection of subsets since we can always append an infiniteMathworldPlanetmathPlanetmath sequence of empty setsMathworldPlanetmath to such a collection.


  • 1 A. Mukherjea, K. Pothoven, Real and Functional analysisPlanetmathPlanetmath, Plenum press, 1978.
  • 2 A. Friedman, Foundations of Modern AnalysisMathworldPlanetmath, Dover publications, 1982.
  • 3 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title outer measure
Canonical name OuterMeasure
Date of creation 2013-03-22 13:45:20
Last modified on 2013-03-22 13:45:20
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Definition
Classification msc 60A10
Classification msc 28A10
Related topic CaratheodorysExtensionTheorem
Related topic CaratheodorysLemma
Related topic ProofOfCaratheodorysExtensionTheorem