On vector measures with values in
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.
