Carath\'eodory's extension theoremCaratheodorysExtensionTheorem2008-11-23 01:12:492008-11-29 22:30:24Theoremexistence of the Lebesgue measuremeasurealgebra of sets$\sigma$-algebra% this is the default PlanetMath preamble. as your knowledge
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\PMlinkescapeword{countably additive}
In measure theory, Carath\'eodory'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 \PMlinkname{countably additive}{Additive} set function on an algebra of sets can be extended to a measure on the \PMlinkname{$\sigma$-algebra}{SigmaAlgebra} generated by that algebra.
\begin{theorem}[Carath\'eodory]
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 $A$.
\end{theorem}
\begin{thebibliography}{9}
\bibitem{williams}
David Williams, \emph{Probability with martingales},
Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
\bibitem{kallenberg}
Olav Kallenberg, \emph{Foundations of modern probability}, Second edition. Probability and its Applications. Springer-Verlag, 2002.
\end{thebibliography}
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