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 -algebra (http://planetmath.org/SigmaAlgebra) generated by that algebra.
Let be a set, be an algebra on , and be the -algebra generated by . If is a countably additive map then there exists a measure on such that on .
- 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|
|Date of creation||2013-03-22 18:33:00|
|Last modified on||2013-03-22 18:33:00|
|Last modified by||gel (22282)|