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

Abstract

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 ( L 1 [ 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 ( Σ i ) , we describe the space of Δ-convergent integrably bounded cf ( X ) -valued martingales in terms of the Δ-closure of ⋃ i = 1 ∞ L 1 [ Σ i , cf ( X ) ] in L 1 [ 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.