# Carathéodory’s extension theorem

In measure theory, Carathéodory’s extension theorem is an important result used in the construction of measures, such as the Lebesgue measure on the real number line. The result states that a countably additive (http://planetmath.org/Additive) set function on an algebra of sets can be extended to a measure on the $\sigma$-algebra (http://planetmath.org/SigmaAlgebra) generated by that algebra.

###### Theorem (Carathéodory).

Let $X$ be a set, $A$ be an algebra on $X$, and $\mathcal{A}\equiv\sigma(A)$ be the $\sigma$-algebra generated by $A$. If $\mu_{0}\colon A\rightarrow\mathbb{R}_{+}\cup\{\infty\}$ is a countably additive map then there exists a measure $\mu$ on $(X,\mathcal{A})$ such that $\mu=\mu_{0}$ on $A$.

## References

• 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 CaratheodorysExtensionTheorem 2013-03-22 18:33:00 2013-03-22 18:33:00 gel (22282) gel (22282) 18 gel (22282) Theorem msc 28A12 Measure OuterMeasure2 LebesgueMeasure CaratheodorysLemma ExistenceOfTheLebesgueMeasure