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Positivity

, Volume 14, Issue 4, pp 715–729 | Cite as

The dual space of L p of a vector measure

  • F. Galaz-Fontes
Article

Abstract

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L p (ν) ×  = E q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L 1 (ν) ×  may be equal or not to E (μ).

Keywords

Vector measure Function norm Associate space σ-Order continuity Fatou property 

Mathematics Subject Classification (2000)

46G10 46E30 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Centro de Investigación en MatemáticasGuanajuato Gto.México

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