Carathéodory’s extension theorem

In measure theory, Carathéodory’s extension theorem is an important result used in the construction of measuresMathworldPlanetmath, such as the Lebesgue measureMathworldPlanetmath on the real number line. The result states that a countably additive ( set functionMathworldPlanetmath on an algebra of setsMathworldPlanetmath can be extended to a measure on the σ-algebra ( generated by that algebra.

Theorem (Carathéodory).

Let X be a set, A be an algebra on X, and Aσ(A) be the σ-algebra generated by A. If μ0:AR+{} is a countably additive map then there exists a measure μ on (X,A) such that μ=μ0 on A.


  • 1 David Williams, Probability with martingales, 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 extension theorem
Canonical name CaratheodorysExtensionTheorem
Date of creation 2013-03-22 18:33:00
Last modified on 2013-03-22 18:33:00
Owner gel (22282)
Last modified by gel (22282)
Numerical id 18
Author gel (22282)
Entry type Theorem
Classification msc 28A12
Related topic Measure
Related topic OuterMeasure2
Related topic LebesgueMeasure
Related topic CaratheodorysLemma
Related topic ExistenceOfTheLebesgueMeasure