Vector Measures, $c_0$, and (sb) Operators
Volume 54 / 2006
Abstract
Emmanuele showed that if ${\mit\Sigma}$ is a $\sigma$-algebra of sets, $X$ is a Banach space, and $\mu : {\mit\Sigma} \to X$ is countably additive with finite variation, then $\mu ({\mit\Sigma} )$ is a Dunford–Pettis set. An extension of this theorem to the setting of bounded and finitely additive vector measures is established. A new characterization of strongly bounded operators on abstract continuous function spaces is given. This characterization motivates the study of the set of (sb) operators. This class of maps is used to extend results of P. Saab dealing with unconditionally converging operators. A characterization of the existence of a countably additive, non-strongly bounded representing measure in terms of $c_0$ is presented. This characterization resolves a question posed in 1970.
