## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 19, Number 1 (2012), 63-80.

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

E. Jiménez Fernández and E. A. Sánchez Pérez

#### 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.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 1 (2012), 63-80.

**Dates**

First available in Project Euclid: 7 March 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1331153409

**Digital Object Identifier**

doi:10.36045/bbms/1331153409

**Mathematical Reviews number (MathSciNet)**

MR2952796

**Zentralblatt MATH identifier**

1247.46034

**Subjects**

Primary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

**Keywords**

Weak orthogonal sequences vector measure integration almost everywhere convergence

#### Citation

Jiménez Fernández, E.; Sánchez Pérez, E. A. Weak orthogonal sequences in $L^2$ of a vector measure and the Menchoff-Rademacher Theorem. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 1, 63--80. doi:10.36045/bbms/1331153409. https://projecteuclid.org/euclid.bbms/1331153409