On vector measures with values in ${\ell }_{\infty }$

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 $W\subseteq {\ell }_{\infty }^{\ast }$ 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 $x^{\ast }\circ \nu$ is countably additive for every $x^{\ast }\in W$. 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 $E^{\ast }$ 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 $L_1(\nu {)}^{\ast }$ is order continuous.