## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 19, Number 2 (2012), 239-256.

### Representation of Banach lattices as $L_w^1$ spaces of a vector measure defined on a $\delta$-ring

O. Delgado and M. A. Juan

#### Abstract

In this paper we prove that every Banach lattice having the Fatou property and having its $\sigma$-order continuous part as an order dense subset, can be represented as the space $L_w^1(\nu)$ of weakly integrable functions with respect to some vector measure $\nu$ defined on a $\delta$-ring.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 2 (2012), 239-256.

**Dates**

First available in Project Euclid: 24 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1337864270

**Digital Object Identifier**

doi:10.36045/bbms/1337864270

**Mathematical Reviews number (MathSciNet)**

MR2977229

**Zentralblatt MATH identifier**

1254.46045

**Subjects**

Primary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]

Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B42: Banach lattices [See also 46A40, 46B40]

**Keywords**

Banach lattice $\delta$-ring Fatou property Order density Order continuity Integration with respect to vector measures

#### Citation

Delgado, O.; Juan, M. A. Representation of Banach lattices as $L_w^1$ spaces of a vector measure defined on a $\delta$-ring. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 2, 239--256. doi:10.36045/bbms/1337864270. https://projecteuclid.org/euclid.bbms/1337864270