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

Abstract

Consider a positive Banach lattice valued vector measure $\bf m:\Sigma \to X$, its space of 2-integrable functions $L^2(\bf m)$ and a sequence $S$ in it. We analyze the notion of weak $\bf m$-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 $0 \le x' \in X'$ such that $S$ is orthogonal with respect to the scalar positive measure $\langle \bf m, x' \rangle$. As an application, we use the representation of $\ell-$sums of $L^2$-spaces as spaces $L^2 (\bf m)$ for a suitable vector measure $\bf m$ centering our attention in the case of $c_0$-sums.