Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki¹

¹ Institute of Mathematics Polish Academy of Sciences Wroc/law Branch Kopernika 18 51-617 Wroc/law, Poland

Studia Mathematica, Tome 165 (2004) no. 2, pp. 111-124

Résumé

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.

Détail

DOI : 10.4064/sm165-2-2

Keywords: positive measure sigma algebra mit sigma subsets set banach space denote mathop nolimits mit sigma banach space x valued measures mit sigma equipped uniform norm mathop nolimits mit sigma its closed subspace consisting those measures which vanish every null set concerned subsets mathcal mathcal mathop nolimits mit sigma defined conditions varphi varphi geq respectively where varphi stands variation varphi mathop nolimits mit sigma establish necessary sufficient conditions mathcal resp mathcal dense mathop nolimits mit sigma resp mathop nolimits mit sigma mathcal mathcal always delta sets establish necessary sufficient conditions sigma sets respective spaces