Characterizations of Discrete Probability Distributions by the Existence of Regular Conditional Distributions Respectively Continuity from Below of Inner Probability Measures

Abstract

Let (Ω, B, P) denote some probability space, where : stands for some Polish space with B as the corresponding Borel σ-algebra. Furthermore, (Ω,B_P,P) is introduced as the completion of (Ω,B,P). It is proved that P is discrete if and only if there exists a regular version of the conditional distribution P(A\B), A ∈ B_p. It follows as a corollary that the x03C3;-algebra consisting of the universally measurable subsets of Ω is not countably generated if and only if Ω is not countable. Furthermore it is shown that P is discrete if and only if the corresponding inner probability measure P_* is continuous from below.