On the [Wikipedia page](https://en.wikipedia.org/wiki/Disintegration_theorem) there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to conditional expectation. Rigorously, how are the two related?



Disintegration, conditional probabilities, and conditional expectation
There's a nice discussion of these issues in "Conditionng as Disintegration" by Chang & Pollard: <https://onlinelibrary.wiley.com/doi/full/10.1111/1467-9574.00056>


