Weak orthogonal sequences in L^2 of a vector measure and the Menchoff -Rademacher Theorem

Abstract

[EN] Consider a positive Banach lattice valued vector measure m : Σ → X, its space of 2-integrable functions L2 (m) and a sequence S in it. We analyze the notion of weak m-orthogonality for such an S in these spaces and we prove a Menchoff-Rademacher Theorem on the almost everywhere convergence of series in them. In order to do this, we provide a criterion for determining when there is a functional 0 ≤ x′ ∈ X′ such that S is orthogonal with respect to the scalar positive measure (m, x′ ). As an application, we use the representation of ℓ−sums of L2-spaces as spaces L2 (m) for a suitable vectormeasure m centering our attention in the case of c0-sums.