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march 2012 Weak orthogonal sequences in $L^2$ of a vector measure and the Menchoff-Rademacher Theorem
E. Jiménez Fernández, E. A. Sánchez Pérez
Bull. Belg. Math. Soc. Simon Stevin 19(1): 63-80 (march 2012). DOI: 10.36045/bbms/1331153409

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.

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E. Jiménez Fernández. E. A. Sánchez Pérez. "Weak orthogonal sequences in $L^2$ of a vector measure and the Menchoff-Rademacher Theorem." Bull. Belg. Math. Soc. Simon Stevin 19 (1) 63 - 80, march 2012. https://doi.org/10.36045/bbms/1331153409

Information

Published: march 2012
First available in Project Euclid: 7 March 2012

zbMATH: 1247.46034
MathSciNet: MR2952796
Digital Object Identifier: 10.36045/bbms/1331153409

Subjects:
Primary: 46E30 , 46G10

Keywords: Almost everywhere convergence , Integration , vector measure , Weak orthogonal sequences

Rights: Copyright © 2012 The Belgian Mathematical Society

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Vol.19 • No. 1 • march 2012
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