Convergence in Law of Random Elements and Random Sets

Abstract

Throughout this paper, we let T denote a fixed topological space (not necessarily Hausdorff) and we let (Ω, F, P) and (ϒ, A, Q) denote two fixed probability spaces. We let (E,E*,E *) denote the P-expectation, the upper and the lower P-expectation. Similarly, we let (E,E*,E*) denote the Q-expectation, the upper and the lower Q-expectation. Recall that a T-valued random element on (Ω, F, P) or (ϒ,A,Q) is simply a map from Ω or ϒ into T. We let C(T) denote the set of all bounded continuous functions o : T → R and we let Ba(T) and B(T) denote the Baire and Borel σ-algebra; that is, Ba(T) is the smallest σ-algebra making all functions in C(T) measurable and B(T) is the smallest σ-algebra containing all closed (or all open) subsets of T. A random element Z on (ϒ,A,Q) is called Baire (Borel) Q-measurable if Z −1(B) is Q-measurable for every Baire (Borel) set B ⊆ T. Similarly, if μ, is a probability measure on some σ-algebra on T, we say that μ is Baireian (Borelian) if every Baire (Borel) subset of T is μ-measurable