Talk:Carathéodory's extension theorem

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I added a small stub on Caratheodoy's extension theorem. This is a fundamental theorem, but unfortunately, I don't know much about topology and measure theory outside the probabilistic context; I am sure there are some links with those fields I cannot do myself. Ashigabou 03:27, 11 February 2006 (UTC)

[edit] Uniqueness

In measure theory, Carathéodory's extension theorem proves that
for a given set Ω, you can always extend a measure defined on R
to the σ-algebra generated by R, where R is a ring included in
the power set of Ω;

In contrast to the extension of a measure on a semi-ring to a measure on a ring, the above requires that there exist a monotonically increasing sequence (A_n)_{n \in \mathbb{N}}, such that \Omega = \bigcup_{n = 1}^\infty A_n. Every subset of Ω can then be covered by a countable union of sets, which is required to define the outer measure that will subsequently be reduced to the desired measure on the generated σ-Algebra.

moreover, the extension is unique.

The extension to a σ-Algebra is unique only if the measure on the generating ring is σ-finite. Note that this already implies the condition above. I therefore suggest to simply add the σ-finiteness to the conditions of the theorem. --Drizzd 10:35, 8 February 2007 (UTC)

Since there were no objections I added the suggested condition to the article. --Drizzd 12:31, 19 February 2007 (UTC)
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