The Existence of Regular Conditional Probabilities: Necessary and Sufficient Conditions

Abstract

Different necessary and sufficient conditions for the existence of regular conditional probabilities are found for the cases of countably generated, countably separated, and complete probability spaces. Perfection is $n.$ and $s.$ for countably generated spaces, "almost pre-standardness" for the countably generated and countably separated cases, and discreteness for complete spaces. Several different forms of the regular conditional probability property must be distinguished.