# Kolmogorov Extension Theorem vs. Caratheodory Extension Theorem

I have mentioned, that CET together with a monotone-class arguments is commonly used in the theory of discrete-time stochastic processes to construct a probability measure out of finite-dimensional distributions. I found it very handy and useful myself, and I guess it extends well to the stochastic fields. At the same time, I often see KET used to construct a probability measure out of finite-dimensional distributions in continuous time case. As it requires the state space to have some topological properties, its application are seems to be less than that of CET.

Updated: more explicit question

I decided to rephrase my questions in a more explicit way: is it true that KET holds without assumptions on the topology of the state space? I have not found the place, where they are used.

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@Martin: I see now, thanks anyway! –  Ilya Jan 2 '13 at 13:56

This is the same answer I gave on MO:

The KET fails for general measurable spaces, the classical example can be found in a paper by Andersen and Jessen. Topological assumptions are necessary so that the resulting measure is not only finitely additive but countably additive. There exists a quasi-topological condition of measure spaces, perfectness, that is sufficient. A probability space $(\Omega,\sigma,\mu)$ is perfect if for every random variable $f:\Omega\to\mathbb{R}$, there exists a Borel set $B\subseteq f(\Omega)$ with measure one under the distribution $\mu\circ f^{-1}$. A proof of KET under the assumption that the marginal measures are perfect due to Lamb is given here. The strategy of the proof is to employ an existence result for regular conditional probability spaces and the construct the proces for them using the Ionescu-Tulcea theorem.

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The point of countable additivity was something that I thought about, but didn't find any topological reasoning when reading the proof. Iounescu-Tulcea also requires topological conditions and does not hold for general spaces (if I'm not mistaken), however Revuz in "Markov Chains" gives a construction of the path space measure without any assumptions on the state space. Maybe whenever you have time, we can talk about this in chat - just ping me, please. –  Ilya Jan 17 '13 at 19:49

Hi here are my two cents,

As far as I know KET's proof use CET (check Karatzas and Sheve's book on Brownian Motion and Stochastic Calculus), and yes it has topological hypothesis (maybe a proof not using CET is possible).

There is another theorem in the same spirit but not using topological hypothesis which is Tulcea Extension Theorem where the topological hytpothesis is replaced by a condition from measure theory which is a bit more satisfying if you want to be more self consistent (i.e. not using topological argument in a "measure theory" theorem).

I think that there is a probability online course by Shalizi where this theorem is at least stated maybe with reference to a proof or even a proof.

Best regards

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Thanks for the two cents :) I know Ionescu-Tulcea theorem: it is commonly used for controlled stochastic processes as it allows using somewhat "natural" conditioning. The proof can be easily found in the appendix of Ethier and Kurtz (or even Bertsekas and Shreve shall contain it, I guess) - however it seems to me a consequence of CET, not an easy/direct one though. –  Ilya Dec 31 '12 at 11:16