Abstract
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 re-
spect to the scalar positive measure m, x′ . As an application, we use the
representation of l2;sums of L2 -spaces as spaces L2 (m) for a suitable vector
measure m centering our attention in the case of c0 -sums.
