Radon-Nikodým derivatives for Banach lattice-valued measures

Abstract:

Let $(\Delta ,\Gamma ,\mu )$ be a measure space such that $0 < \mu (\Delta ) < \infty$ and such that $\Gamma$ has no $\mu$-atoms. Furthermore, let $E$ be a Dedekind complete Banach lattice. By $M(\mu ,E)$ we denote the set of all $E$-valued set functions $\nu$ on $\Gamma$ satisfying (i) $\nu$ is additive, (ii) $\nu$ is order bounded and of bounded variation, (iii) $\nu$ is $\mu$-absolutely continuous (with respect to the norm topology on $E$), (iv) ${\nu ^ + }$ and ${\nu ^ - }$ satisfy \[ \inf \{ \sup \{ {\upsilon ^{ + / - }}(A):\mu (A) < \varepsilon \} :\varepsilon > 0\} = 0.\] By ${L_1}(\mu ,E)$ we denote the set of $E$-valued Bochner integrable functions on $\Delta$. It is shown that under the canonical map \[ f \mapsto {\nu _f}\] (where ${\nu _f}(A) = \smallint f{\mathcal {X}_A}d\mu$, $A \in \Gamma$), ${L_1}(\mu ,E)$ is a Riesz subspace of the Dedekind complete Riesz space $M(\mu ,E)$. Furthermore, the following theorems are proved. Theorem. Equivalent are (a) ${l_\infty }$ is not isomorphic to a closed sublattice of $E$. (b) ${L_1}(\mu ,E)$ is isomorphic to an ideal in $M(\mu ,E)$. Theorem. Equivalent are (a) ${c_0}$ is not isomorphic to a closed sublattice of $E$. (b) ${L_1}(\mu ,E)$ is isomorphic to $M(\mu ,E)$.