# Associate space with respect to a locally $$\sigma$$ σ -finite measure on a $$\delta$$ δ -ring and applications to spaces of integrable functions defined by a vector measure

Associate space with respect to a locally $$\sigma$$ σ -finite measure on a $$\delta$$... 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 functions defined by a vector measure

, Volume 20 (3) – Sep 26, 2015
25 pages

/lp/springer_journal/associate-space-with-respect-to-a-locally-sigma-finite-measure-on-a-Zsq625g2x0
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0370-4
Publisher site
See Article on Publisher Site

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