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Associate space with respect to a locally $$\sigma $$ σ -finite measure on a $$\delta $$ δ -ring and applications to spaces of integrable fu ...

Associate space with respect to a locally $$\sigma $$ σ -finite measure on a $$\delta $$ δ -ring... We show that for a locally $$\sigma $$ σ -finite measure $$\mu $$ μ defined on a $$\delta $$ δ -ring, the associate space theory can be developed as in the $$\sigma $$ σ -finite case, and corresponding properties are obtained. Given a saturated $$\sigma $$ σ -order continuous $$\mu $$ μ -Banach function space E, we prove that its dual space can be identified with the associate space $$E ^\times $$ E × if, and only if, $$E^\times $$ E × has the Fatou property. Applying the theory to the spaces $$L^p (\nu )$$ L p ( ν ) and $$L_w^p (\nu )$$ L w p ( ν ) , where $$\nu $$ ν is a vector measure defined on a $$\delta $$ δ -ring $$\mathcal {R}$$ R and $$1 \le p < \infty $$ 1 ≤ p < ∞ , we establish results corresponding to those of the case when the vector measure is defined on a $$\sigma $$ σ -algebra. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Associate space with respect to a locally $$\sigma $$ σ -finite measure on a $$\delta $$ δ -ring and applications to spaces of integrable fu ...

Avalos-Ramos, Celia; Galaz-Fontes, Fernando
Positivity , Volume 20 (3) – Sep 26, 2015
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References (32)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
DOI
10.1007/s11117-015-0370-4
Publisher site
See Article on Publisher Site

Abstract

We show that for a locally $$\sigma $$ σ -finite measure $$\mu $$ μ defined on a $$\delta $$ δ -ring, the associate space theory can be developed as in the $$\sigma $$ σ -finite case, and corresponding properties are obtained. Given a saturated $$\sigma $$ σ -order continuous $$\mu $$ μ -Banach function space E, we prove that its dual space can be identified with the associate space $$E ^\times $$ E × if, and only if, $$E^\times $$ E × has the Fatou property. Applying the theory to the spaces $$L^p (\nu )$$ L p ( ν ) and $$L_w^p (\nu )$$ L w p ( ν ) , where $$\nu $$ ν is a vector measure defined on a $$\delta $$ δ -ring $$\mathcal {R}$$ R and $$1 \le p < \infty $$ 1 ≤ p < ∞ , we establish results corresponding to those of the case when the vector measure is defined on a $$\sigma $$ σ -algebra.

Journal

Positivity – Springer Journals

Published: Sep 26, 2015

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