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Carathéodory's extension theorem
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(Theorem)
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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 set function on an algebra of sets can be extended to a measure on the $\sigma$ -algebra generated by that algebra.
Theorem 1 (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 $A$ .
- 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.
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"Carathéodory's extension theorem" is owned by gel.
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Cross-references: map, algebra, generated by, algebra of sets, set function, line, real number, Lebesgue measure, theory, measure
There is 1 reference to this entry.
This is version 15 of Carathéodory's extension theorem, born on 2008-11-23, modified 2008-11-29.
Object id is 11270, canonical name is CaratheodorysExtensionTheorem.
Accessed 1028 times total.
Classification:
| AMS MSC: | 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities) |
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Pending Errata and Addenda
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