A description of norm-convergent martingales on vector-valued Lp-spaces

Norm-convergent martingales on tensor products of Banach spaces are considered in a measure-free setting. As a consequence, we obtain the following characterization for convergent martingales on vectorvalued Lp-spaces: Let (Ω,Σ,μ) be a probability space, X a Banach space and (Σn) an increasing sequence of sub σ-algebras of Σ. In order for (fn,Σn)∞n=1 to be a convergent martingale in Lp(μ,X) (1 p <∞) it is necessary and sufficient that, for each i ∈ N, there exists a convergent martingale (x (n) i ,Σn)∞n=1 in Lp(μ) and yi ∈ X such that, for each n ∈ N, we have fn(s) = ∞ i=1 x (n) i (s)yi for all s ∈ Ω, where ∞i =1 |limn→∞x (n) i | Lp(μ) <∞and limi→∞ yi →0. © 2005 Elsevier Inc. All rights reserved.