Talk:Hahn–Kolmogorov theorem

Following "Real Analysis" by Gerald B. Folland, the extended measure is unique if the premeasure is sigma-finite, but not necessarily if it is not sigma-finite. In the general case one can only show something that is rather close to uniqueness. I don't know any counterexamples, though.

Algebra of Sets
I changed the target of this link to 'Sigma-algebra' b/c it seemed to be an explanation better suited for this article. Zero sharp 05:03, 29 October 2006 (UTC)

I believe the "Hahn-Kolmogorov" theorem as stated, without the assumption that \mu_0 is non negative is actually false. For instance, take X = {1,2,3,....} U { w }, i.e. the natural numbers N plus an extra symbol w. Let F the collection of finite subsets of N and G be the subsets of the form M U {w} where M is a subset of N such that N - M is finite. The collection Sigma_0 = F U G is an Algebra because X = N U {w} in G,  F U F = F,  G U Sigma_0 = G, X - F \subset G and X - G \subset F. Let \mu_0 be given by \mu_0(g) = +\infty for g in G and \mu_0(f) = sum_{i in f} (-1)^i for f in F. \mu_0 satisfies the hypothesis of the theorem stated in Wikipedia but there is no sigma-additive extension to sigma(A) = Parts of X, because it is not possible to assign a value to \mu(N).

If what I say above is correct then the "Hahn-Kolmogorov" page should be simply removed from the Wikipedia, because by adding the hypothesis that \mu_0 is non-negative we get what is widely known as the Caratheodory extension theorem, and this seems to be the correct reference. Walterfm —Preceding unsigned comment added by Walterfm (talk • contribs) 17:42, 25 June 2010 (UTC)


 * Answer to Walterfm's remark:

You are right: in order to the result be valid we need to assume $$ \mu_0 $$ to be non-negative. HOWEVER there is not exactly a widely known Carathéodory Theorem! Some authors (such as Tao) presents Carathéodory Theorem just as the construction of a $$ \sigma $$-ring and a measure from an outer-measure, and the Hahn-Kolmogorov Theorem as how, using Carathéodory Theorem, one can actually build extensions. Some other author, puts everything under the name "Carathéodory Theorem" (yet, some other authors (such Halmos) just present the theorems without calling them nor "Hahn-Kolmogorov" nor "Carathéodory").

I agree the Wikipedia article should make explicit that there are those naming divergences.

Actually, Hahn-Komorogorov Theorem is a stronger theorem than Caratheodory extension theorem, right? (I am using the Wikipedia's naming scheme here.) It looks to me that Caratheodory's Theorem is just HK Theorem with "ring" replaced with "algebra". And by definition, all rings are algebras. So HK implies Caratheodory. Kelvinator0

Uniqueness
If $$\mu_0$$ is not $$\sigma$$-finite then the extension need not be unique, even if the extension itself is $$\sigma$$-finite. See, for instance, Halmos, Measure Theory, sections 13, exercise 5.

Here is an example (based on the abovementioned exercise).

We call rational closed-open interval, any subset of $$\mathbb{Q}$$ of the form $$[a,b)$$, where $$a, b \in \mathbb{Q}$$.

Let $$X$$ be $$\mathbb{Q}\cap[0,1)$$ and let $$\Sigma_0$$ be the algebra of all finite union of rational closed-open intervals contained in $$\mathbb{Q}\cap[0,1)$$. It is easy to prove that $$\Sigma_0$$ is, in fact, an algebra. It is also easy to see that every non-empty set in $$\Sigma_0$$ is infinite.

Let $$\mu_0$$ be the counting set function ($$\#$$) defined in $$\Sigma_0$$. It is clear that $$\mu_0$$ is finitely additive and $$\sigma$$-additive in $$\Sigma_0$$. Since every non-empty set in $$\Sigma_0$$ is infinite, we have, for every non-empty set $$A\in\Sigma_0$$, $$\mu_0(A)=+\infty$$

Now, let $$\Sigma$$ be the $$\sigma$$-algebra generated by $$\Sigma_0$$. It is easy to see that $$\Sigma$$ is the set of all subset of $$X$$, and both $$\#$$ and $$2.\#$$ are $$\sigma$$-finite measures defined on $$\Sigma$$ and both are extensions of $$\mu_0$$.

Attribution: Hahn–Kolmogorov 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) 16:28, 9 November 2013 (UTC)


 * The article on the Caratheodory and Hahn-Kolmogorov theorems should be merged. 129.215.104.198 (talk) 18:08, 9 October 2015 (UTC)


 * That is what I was going to say. This is the same theorem (or set of theorems, with variations on the statement: algebra, ring, semiring…) under different names. Currently the article “Carathéodory extension theorem” subsumes this one, except its example.
 * Maëlan 22:50, 25 June 2017 (UTC)


 * Yes, please get rid of this article. --Johnmichaelwu (talk) 20:54, 6 October 2019 (UTC)


 * Ugh. I'll attempt a merge now. 67.198.37.16 (talk) 20:54, 3 October 2020 (UTC)