# Disintegration theorem, a reference needed

I've come a cross the so called 'disintegration theorem' in multiple occasions now and I'm interested to learn its proof and more on related topics. Particularly I'm interested in a proof for the formulation presented by Wikipedia http://en.wikipedia.org/wiki/Disintegration_theorem .

However, the reference that Wikipedia provides is a very old book (from the 70's with no fancy latex styling) and I can't find other than the oldest version of it. Most of the notations embed very unreadably and awkwardly to the text that I haven't succeeded to follow his arguments so far (could be just unpatiency) and I'm looking for another source that is more readable. If someone has a newer textbook covering this topic then I would be glad to look it up and study this subject.

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Isn't this equivalent to the existence of regular conditional probabilities? – Stefan Hansen Jul 27 '12 at 16:32
possible duplicate of Conditional probability and the disintegration theorem – Did Jul 27 '12 at 21:07
@StefanHansen: Could be, I'm not that familiar with conditional probability theory. If you can point some sources then I would be glad to go them through. And for others: I'm also looking for sources that deal with general metric spaces (locally compact Polish, compact?) with the push-forward framework and disintegration. Mainly in non probabilistic setting. – T. Eskin Jul 28 '12 at 20:55
About general metric spaces: did you open at least once Billingsley's book mentioned in my answer? – Did Jul 31 '12 at 7:50
I would like to raise a level of (humorous) awareness that books from the 70's are now considered "really old." – Emily Jul 31 '12 at 15:56

The most comprehensive account of disintegration can be found in David Fremlin's magnum opus Measure Theory in Chapter 45 in Volume 4.1. But this is certainly overkill.

The state of the art paper is Disintegration and Compact Measures by Jan Pachl, he gives a characterization result and nothing more general is possible. His approach is based on the von Neumann-Maharam lifting theorem and this approach to disintegrations was pioneered by Hoffmann-Jorgensen in Existence of Conditional Probabilities. The approach based on liftings has the advantage that it needs no separability conditions, the cost is that the disintegrations are only measurable with respect to a completion.

Under separability assumptions, there are necessary and sufficient conditions known for certain kinds of disintegrations. You can find very useful results to this effect in a beautiful paper by Arnold Faden: The Existence of Regular Conditional Probabilities: Necessary and Sufficient Conditions.

Now these paper represent the high end of mathematical probability theory. For most applications, one can use much more elementary methods. For conditioning on $\mathbb{R}$, the book by Lehmann and Romano already mentioned gives a very readable proof under Theorem 2.5.1 (2005 ed.). In Billingsley's, Probability and Measure, 3rd ed, you can find the same result in section 33 as Theorem 33.3. This results are more powerful than it may seem at first. By a famous isomorphism theorem, every uncountable, separable and complete metric space endowed with the Borel $\sigma$-algebra is isomorphic as a measurable space to $\mathbb{R}$ with the Borel $\sigma$-algebra. If you want to see a proof of the strong version of the result without using this isomorphism theorem, you can check Theorem 10.2.2 in Dudley's Real Analysis and Probability (2002/2004 version). The proof in Dudley is considerably harder.

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So far Fremlin's book has been the most impressive and it is exactly what I was looking for. Not only is this opus amazing but so are his other books of the same 'series'. I also looked up rest of your suggestions and they were useful too, thank you. The bounty can be awarded after 1 hour, so until then if nothing better (which I doubt) appears then it surely belongs to this answer. – T. Eskin Aug 1 '12 at 13:54
I don't usually edit other people's posts, so please note that the link to Pachl's aritcle has become invalid. A new one is mscand.dk/article/view/11771/9787. Also, his theorem only proves existence, without concern for uniqueness. – Alex M. Jul 12 at 15:06
@AlexM. Thank you forpointing the link rot out. I think outside the coutably generated case, uniqueness is too much to hope for. – Michael Greinecker Jul 13 at 6:11

Chapter 6 of Foundations of Modern Probability (Second edition) by Olav Kallenberg starts with the sentence

"Modern probability theory can be said to begin with the notions of conditioning and disintegration."

I think you will find the whole chapter a useful reference.

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From the other page:

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A nice book on Measure Theory by Bogachev deals with disintegration in Chapter 10 (Volume 2).

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Pollard's book "A User's Guide to Measure Theoretic Probability" is very insightful in regard to relating disintegration and conditional probabilities.

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