Talk:Carathéodory's extension theorem

Stub
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)

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 $$\Omega$$ 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 $$\sigma$$-Algebra.

moreover, the extension is unique.

The extension to a $$\sigma$$-Algebra is unique only if the measure on the generating ring is $$\sigma$$-finite. Note that this already implies the condition above. I therefore suggest to simply add the $$\sigma$$-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)

Attribution: Carathéodory or not?
"Warning: I've seen the following theorem called the Carathéodory extension theorem, the Carathéodory-Fréchet extension theorem, the Carathéodory-Hopf extension theorem, the Hopf extension theorem, the Hahn-Kolmogorov extension theorem, and many others that I can't remember! We shall simply call it Extension Theorem. However, I read in Folland's book (p. 41) that the theorem is originally due to Maurice René Fréchet (1878–1973) who proved it in 1924." Paul Loya (page 33). Boris Tsirelson (talk) 16:15, 9 November 2013 (UTC)


 * The article on the Caratheodory and Hahn-Kolmogorov theorems should be merged. 129.215.104.198 (talk) 18:07, 9 October 2015 (UTC)


 * I have just now merged from Hahn-Kolmogorov theorem just now. The talk page there listed four different people stating it should be merged, plus the statement above makes five, plus my own reading makes six so that is six votes to merge. For other comments, refer to Talk:Hahn-Kolmogorov theorem. In particular, there is some burblings as to which variants are more general than others: Is the statement using semi-rings more general, or less, than the one for field of sets? I can't quite tell, off-hand. 67.198.37.16 (talk) 02:28, 4 October 2020 (UTC)

σ-finiteness of the extension
It is claimed that

"If μ is σ-finite then the extension μ′ is unique (and also σ-finite)."

Unless I have made a mistake, the claim inside parentheses is false. Consider the counting measure μ defined on the ring R of finite subsets of the real line. It is finite (and hence σ-finite), but its extension μ′ is not, since the real line (which is in the σ-field generated by R) is not a countable union of finite sets.

The extension to the σ-ring generated by R is σ-finite, however (see, e.g., Theorem 2.5.3 from "A basic course in measure and probability: theory for applications", by Ross Leadbetter, Stamatis Cambanis and Vladas Pipiras). In particular, if the whole space is a countable union of elements from the ring, then μ′ is σ-finite.

Note that the last condition (actually, something apparently slightly stronger, since the union is required to be disjoint) is required under the alternative definition for rings given in the first section of this article, but it doesn't seem to be made explicit that such condition is needed if we want to conclude μ′ is σ-finite. — Preceding unsigned comment added by TuringMachine001 (talk • contribs) 15:44, 15 February 2016 (UTC)

Examples of non-uniqueness of extension - Via Fubini's theorem
The second measure presented is:

The measure of a subset is $$\int_0^1n(x)dx$$ where $$n(x)$$ is the number of points of the subset with given $$x$$-coordinate.

However, since $$Y$$ is the unit interval with the discrete counting measure, it is not $$\sigma$$-finite. So, we cannot conclude that the function $$x \mapsto n(x) $$ is measurable. If it is not measurable, then $$\int_0^1n(x)dx$$ is not defined.

Example: Let $$C=[0,1]\times E$$, where $$E$$ is any non-Lebesgue measurable subset. Clearly $$C \in R$$ (because $$R$$ is the ring generated by products $$A\times B$$ where $$A$$ is Lebesgue measurable and $$B$$ is any subset). Let $$D = \{(x,x) : x \in [0,1] \}$$. It is easy to see that $$D \in \sigma(R)$$. Let $$ S = C \cap D $$. It follows immediately that $$ S \in \sigma(R)$$, $$ S = \{(x,x) : x \in E \} $$ and $$n_S$$ is the indicator function of $$E$$ and so it is not a measurable function.

50.100.95.67 (talk) 05:28, 26 March 2023 (UTC)

Something is wrong with the "Introductory statement"
I do not see why both "finite additivity" and "sigma additivity" are written there.

Sigma additivity should imply finite additivity, just replace all $$\infty$$ with $$N$$. If you really need infinite sequences, then add as many empty sets as you need, as $$\empty$$ is has to be even in a semialgebra, not just algebra.