Carathéodory’s lemma

In measure theory, Carathéodory’s lemma is used for constructing measuresMathworldPlanetmath and, for example, can be applied to the construction of the Lebesgue measureMathworldPlanetmath and is used in the proof of Carathéodory’s extension theorem. The idea is that to define a measure on a measurable spaceMathworldPlanetmathPlanetmath (X,𝒜) we would first construct an outer measureMathworldPlanetmathPlanetmath (, which is a set functionMathworldPlanetmath defined on the power setMathworldPlanetmath of X. Then, this outer measure is restricted to 𝒜 and Carathéodory’s lemma is applied to show that this restrictionPlanetmathPlanetmath does in fact result in a measure. For an example of this procedure, see the proof of Carathéodory’s extension theorem.

Given an outer measure μ on a set X, the result first defines a collectionMathworldPlanetmath of subsets of X — the μ-measurable sets. A subset SX is called μ-measurable (or Carathéodory measurable with respect to μ) if the equality


holds for every EX. Then, Caratheodory’s lemma says that a measure is obtained by restricting μ to the μ-measurable sets.

Lemma (Carathéodory).

Let μ be an outer measure on a set X, and A be the class of μ-measurable sets. Then A is a σ-algebra ( and the restriction of μ to A is a measure.

It should be noted that for any outer measure μ and sets S,EX, subadditivity of μ implies that the inequalityMathworldPlanetmath μ(E)μ(ES)+μ(ESc) is always satisfied. So, only the reverse inequality is required and consequently S is μ-measurable if and only if


for every EX.


  • 1 David Williams, Probability with martingalesMathworldPlanetmath, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
  • 2 Olav Kallenberg, Foundations of modern probability, Second edition. Probability and its Applications. Springer-Verlag, 2002.
Title Carathéodory’s lemma
Canonical name CaratheodorysLemma
Date of creation 2013-03-22 18:33:03
Last modified on 2013-03-22 18:33:03
Owner gel (22282)
Last modified by gel (22282)
Numerical id 19
Author gel (22282)
Entry type Theorem
Classification msc 28A12
Related topic CaratheodorysExtensionTheorem
Related topic OuterMeasure2
Related topic LebesgueOuterMeasure
Related topic ConstructionOfOuterMeasures
Related topic ProofOfCaratheodorysLemma
Related topic ProofOfCaratheodorysExtensionTheorem
Defines Carathéodory measurable