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→Relation to conditional expectation: unnecessary comment, not necessarily true. Does not seem to reflect assertion in ref., just reflects order of its TOC |
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In other words <math>\kappa_{Y|X}</math> is a [[Markov kernel]].
The first condition holds trivially, but the proof of the second is more involved{{Dubious?|1=Which regularity condition is trivial|reason=Ain't it the other way around?|date=November 2022}}. It can be shown that if ''Y'' is a random element <math>\Omega \to S</math> in a [[Radon space]] ''S'', there exists a <math>\kappa_{Y|X}</math> that satisfies the measurability condition.<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability theory : a comprehensive course |___location=London |isbn=978-1-4471-5361-0 |edition=Second}}</ref> It is possible to construct more general spaces where a regular conditional probability distribution does not exist.<ref>Faden, A.M., 1985. The existence of regular conditional probabilities: necessary and sufficient conditions. ''The Annals of Probability'', 13(1), pp.288-298.</ref>
=== Relation to conditional expectation ===
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