On semiregular conditional distributions

Abstract

Let (Ω,A,P) denote some probability space and ℬ some sub-σ-algebra ofA. It is shown that there exists a semiregular versionQ _ω(A),A, ε∈Ω, of the conditional distributionP(A|ℬ), A∈A, i.e., ω→Q _ω(A), ω∈Ω (A∈A fixed) is ℬ andA→Q _ω(A),A∈A (ω∈Ω fixed), is a probability charge satisfyingQ _ω(N)=0, ω∈Ω, for allP-zero setsN∈ℬ, if and only ifL ₁(Ω,ℬP|ℬ) has a lifting, which exists for any sub-σ-algebra ℬ ofA ifL ₁(Ω,A P) is separable. Separability ofL ₁(Ω,A,P) implies also the existence of a strongly semiregular versionQ _ω(A),A, ω∈Ω, ofP(A|ℬ), A ∈ℬ, i.e., ω→Q _ω(A), ω∈Ω (A∈A fixed), is ℬ-measurable andA→Q _ω(A),A (ω∈Ω fixed), is a probability charge. Furthermore,P can be written as αP ₁+(1−α)P ₂, 0≤α≤1, whereP ₁ are probability measures onA such thatP ₁(A|ℬ),A∈A, has a semiregular version vanishing for anyP-zero setN∈ℬ andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<α<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ _ω(N)=0 for all ω∈Ω and anyP-zero setN∈ℬ is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.