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 vector-valued 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 (xi(n),Σn)n=1∞ in Lp(μ) and yi∈X such that, for each n∈N, we havefn(s)=∑i=1∞xi(n)(s)yifor all s∈Ω, where ‖∑i=1∞|limn→∞xi(n)|‖Lp(μ)<∞ and limi→∞‖yi‖→0.