Denseness and Borel complexity of some sets of vector measures
Tom 165 / 2004
Streszczenie
Let $\nu $ be a positive measure on a $\sigma $-algebra ${\mit \Sigma }$ of subsets of some set and let $X$ be a Banach space. Denote by $\mathop {\rm ca}\nolimits ({\mit \Sigma },X)$ the Banach space of $X$-valued measures on ${\mit \Sigma }$, equipped with the uniform norm, and by $\mathop {\rm ca}\nolimits ({\mit \Sigma },\nu ,X)$ its closed subspace consisting of those measures which vanish at every $\nu $-null set. We are concerned with the subsets ${\mathcal E}_\nu (X)$ and ${\mathcal A}_\nu (X)$ of $\mathop {\rm ca}\nolimits ({\mit \Sigma },X)$ defined by the conditions $|\varphi |=\nu $ and $|\varphi |\geq \nu $, respectively, where $|\varphi |$ stands for the variation of $\varphi \in \mathop {\rm ca}\nolimits ({\mit \Sigma },X)$. We establish necessary and sufficient conditions that ${\mathcal E}_\nu (X)$ [resp., ${\mathcal A}_\nu (X)$] be dense in $\mathop {\rm ca}\nolimits ({\mit \Sigma },\nu ,X)$ [resp., $\mathop {\rm ca}\nolimits ({\mit \Sigma },X)$]. We also show that ${\mathcal E}_\nu (X)$ and ${\mathcal A}_\nu (X)$ are always $G_\delta $-sets and establish necessary and sufficient conditions that they be $F_\sigma $-sets in the respective spaces.
