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 $\mathrm{A}\mathrm{\equiv }\sigma \mathit{}\mathrm{(}A\mathrm{)}$ be the $\sigma$-algebra generated by $A$. If ${\mu }_{\mathrm{0}}\mathrm{:}A\mathrm{\to }{\mathrm{R}}_{\mathrm{+}}\mathrm{\cup }\mathrm{\{}\mathrm{\infty }\mathrm{\}}$ is a countably additive map then there exists a measure $\mu$ on $\mathrm{(}X\mathrm{,}\mathrm{A}\mathrm{)}$ such that $\mu \mathrm{=}{\mu }_{\mathrm{0}}$ on $A$.

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