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Does the "projective limit" concept exist for probability spaces? The only result that I know of seems to be the Kolmogorov-Daniell theorem, but this is just a particular case where the spaces involved are finite products of one and the same Polish space.

If supplementary conditions are necessary (such as the spaces being topological and the probabilities being inner-regular etc.), then why are they needed? The question is motivated by the fact that the (arbitrary) product can be constructed in a purely measurable context, without any topological assumptions.

Finally, why is the space in the Kolmogorov-Daniell theorem required to be Polish? The proof only uses the fact that finite products of Polish spaces remain Polish and that probabilities on them are inner-regular; but both properties stay valid for an even more general class of spaces (Suslin spaces, for sure; I am not sure about Radon spaces), so why only Polish spaces?

Alex M.
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    I think that the answer to your last question is that that is a peculiarity of the particular formulation of the extension theorem that you saw. Check the statement in Volume II of Bogachev's measure theory for example. The only thing you need for the proof to work is inner regularity (where the usual definition of compactness is replaced with the finite intersection property), which can be done in a purely measure theoretic framework. – Chris Janjigian Aug 02 '14 at 17:39
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    I am hardly in a position to pinpoint exactly why the supplementary conditions are necessary (your second question), but I did run into a counterexample for the general case in the literature. It is covered in exercise 10.6.5 of Cohn's Measure Theory and at the end of §35 in Bauer's Probability Theory. Bauer furthermore refers to The General Marginal Problem, which is a set of lecture notes published by J. Hoffman–Jørgensen, which supposedly provides "an exhaustive treatment of the problem (in a much wider setting)". – Josse van Dobben de Bruyn Jul 18 '17 at 21:05
  • @JossevanDobbendeBruyn: Thank your for your interest. I contacted Hoffman–Jørgensen at the time when I also asked this question, and he was very kind to send me an electronic copy of his book; after having browsed through it, let me just say that it is a VERY difficult read. Anyway, a satisfactory answer is given by a theorem attributed to Prokhorov (a version of it can be found in an old version of a Wikipedia article, that someone removed without explanation). – Alex M. Jul 18 '17 at 21:53
  • @AlexM. Once again, you are way ahead of me. ;-) I didn't really get very far in Hoffman-Jørgensen's lecture notes either, but that may have been due to lack of serious attempt (I was sufficiently put off by the typeface). I'm not sure why the Wikipedia section has been removed, but maybe someone felt the material was not supported by a proper source? In any case, I encourage you to expand your comment to an answer, especially if you can support it with references to the literature! – Josse van Dobben de Bruyn Jul 18 '17 at 22:14
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    I don't think that this question is still active, but for further reference: there are the two papers http://dx.doi.org/10.1112/plms/s3-8.3.321 and https://doi.org/10.1137/1138027 that treat this problem, see also https://math.stackexchange.com/q/2971462/743552. – Mushu Nrek Apr 02 '25 at 10:06

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