Atomic Intersection of s-Fields and Some of Its Consequences

Abstract

Let (omega,F,P) be a probability space. For each G in F, define G as the s-field generated by G and those sets f in F satisfying P(f) in {0, 1}. Conditions for P to be atomic on the intersection of the complements of Ai for i=1,..,k, with A1, . . . ,Ak in F sub-s-fields, are given. Conditions for P to be 0-1-valued on the intersection of the complements of Ai for i=1,..,k are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.Atomic probability measure, Gibbs sampling, Graphical models, Intersection property, Iterated conditional expectations.