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 $A$.
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 |
---|---|
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 |