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.
Also, any kind of discussion about this theorem and its applications are warmly welcome.
Thanks in advance.
 A: A nice book on Measure Theory by Bogachev deals with disintegration in Chapter 10 (Volume 2).
A: 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.
A: 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.
A: From the other page:


*

*Testing Statistical Hypotheses by Lehmann and Romano. 

*Probability and Measure by Billingsley.

A: Pollard's book "A User's Guide to Measure Theoretic Probability" is very insightful in regard to relating disintegration and conditional probabilities.
Here: http://www.stat.yale.edu/~pollard/
A: Check out the paper Graf, S., and Mauldin, R. D. A classification of disintegrations of measures. In Measure and measurable dynamics (Rochester, NY, 1987), no. 94 in Contemp. Math. Amer. Math. Soc., Providence, RI, 1989, pp. 147-158.
This link works.
A: See also Integration and Probability by Paul Malliavin, second edition (1993), Theorem 6.5.1 (page 227).
