On the existence of regular conditional probabilities

Summary Let (X, A, P) be a measure space with P(X)=1 and a sub--algebra. It is well known (see e.g. Doob, p. 624) that even if A is separable, a regular conditional probability (r.c.p.) on A, given , does not always exist. All theorems assuring the existence of a r.c.p. known hitherto use in addition to the separability of A conditions of an essentially topological nature such as compact approximation (for references see e.g. Pfanzagl-Pierlo (1966), section 7).It is the purpose of this paper to explore the possibility of sufficient conditions of a purely measure-theoretic nature. Without loss of generality we may assume that (X, A, P) is complete and is complete with respect to P ¦A. For many applications it will be sufficient to have the r.c.p. defined on a sub--algebra A ₀ A with A ₀ ~ A(P|A) rather than on the whole -algebra A. It will be shown that a r.c.p. in this more restricted sense always exists if A is the completion of a separable -algebra and if it contains a P-null set of the power of the continuum. The P-null-set -condition cannot be omitted without any replacements. To be more specific: Even a r.c.p. in the restricted sense does not necessarily exist if A is the completion of a separable -algebra (see Example 1). Furthermore, there are instances, in which a r.c.p. in the restricted sense exists but a r.c.p. in the usual sense does not exist (see Example 2).Even if a r.c.p. exists, it is not necessarily proper. The deep study by Blackwell and Ryll-Nardzewski (1963) reveals that this negative statement even holds under rather restrictive conditions (X = complete separable metric space, A = Borel algebra of X). It therefore seems to be of some interest to state that the r.c.p. in the restricted sense can always be chosen such that it is proper with respect to some sub--algebra +B_o +(+B A ₀ with ₀₀(P¦A).