# Conditional probability and the disintegration theorem

I was wondering how conditional probability and the disintegration theorem are related?

How is the conditional probability given by the disintegration theorem? I don't quite understand what Wikipedia says:

The disintegration theorem can be applied to give a rigorous treatment of conditioning probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.

The linked paper Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration". STATISTICA NEERLANDICA 51 (3) has 30 pages and is somehow overwhelming to me now.

Following is a scenario I am considering (but it may be over-complicated and misleading, I am not sure):

Suppose $(\Omega, \mathcal{M},P)$ is a probability space. $(S, \mathcal{S})$ is a measurable space. and $Y:\Omega \to S$ is a measurable mapping. The conditional probability $P(\cdot | Y)$ is a mapping $\mathcal{M} \times S \to [0,1]$.

I guess it is only when the conditional probability is regular, i.e. $\forall s \in S$, $P(\cdot | Y)(s)$ is a probability measure, that the conditional probability can be given by the disintegration theorem? Then, how is the conditional probability given by the disintegration theorem?

Note: I think I have got the idea of the disintegration theorem. My source is also Wikipedia.

I also appreciate it if you could let me know about some accessible relevant texts.

Thanks and regards!

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The general topic of conditioning gets complicated very fast, but for practical purposes it is usually enough to work with conditioning on $\mathbb{R}$ with the Borel sets (most spaces one encounters in practice are isomorphic to $\mathbb{R}$ as measurable spaces). The clearest exposition of conditioning in this case I have seen is in the book Testing Statistical Hypotheses by Lehmann and Romano. A close second is Probability and Measure by Billingsley.