On vector measures with values in $\ell_\infty $
Volume 274 / 2024
Abstract
We study some aspects of countably additive vector measures with values in $\ell _\infty $ and the Banach lattices of real-valued functions that are integrable with respect to such a vector measure. On the one hand, we prove that if $W \subseteq \ell _\infty ^*$ is a total set not containing sets equivalent to the canonical basis of $\ell _1 (\mathfrak c)$, then there is a non-countably-additive $\ell _\infty $-valued map $\nu $ defined on a $\sigma $-algebra such that the composition $x^* \circ \nu $ is countably additive for every $x^*\in W$. On the other hand, we show that a Banach lattice $E$ is separable whenever it admits a countable, positively norming set and both $E$ and $E^*$ are order continuous. As a consequence, if $\nu $ is a countably additive vector measure defined on a $\sigma $-algebra and taking values in a separable Banach space, then the space $L_1(\nu )$ is separable whenever $L_1(\nu )^*$ is order continuous.
