For working with <math>\kappa_{Y|X}</math>, it is important that it be ''regular'', that is:
*# For almost all ''x'', <math>A \mapsto \kappa_{Y|X}(x, A)</math> is a probability measure
*# For all ''A'', <math>x \mapsto \kappa_{Y|X}(x, A)</math> is a measurable function
In other words <math>\kappa_{Y|X}</math> is a [[Markov kernel]].
The formerfirst condition is trivialholds trivially, but the proof of the lattersecond is more involved. 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>
