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

Abstract

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