A Banach lattice approach to convergent integrably bounded set-valued martingales and their positive parts

Denote by cf(X) the set of all nonempty convex closed subsets of a separable Banach space X. Let (Ω,Σ,μ) be a complete probability space and denote by (L1[Σ,cf(X)],Δ) the complete metric space of (equivalence classes of a.e. equal) integrably bounded cf(X)-valued functions. For any preassigned filtration (Σi ), we describe the space of Δ-convergent integrably bounded cf(X)-valued martingales in terms of the Δ-closure of ∞ i=1 L1[Σi , cf(X)] in L1[Σ,cf(X)]. In particular, we provide a formula to calculate the join of two such martingales and the positive part of such a martingale. Our object is achieved by considering the more general setting of a near vector lattice (S, d), endowed with a Riesz metric d. By means of Rådström’s embedding theorem for such spaces, a link is established between the space of convergent martingales in S and the space of convergent martingales in the Rådström completion R(S) of S. This link provides information about the former space of martingales, via known properties of measure-free martingales in Riesz normed vector lattices, applicable to R(S). We also apply our general results to the spaces of Δ-convergent ck(X)-valued martingales, where ck(X) denotes the set of all nonempty convex compact subsets of X. © 2007 Published by Elsevier Inc.