Sequential closures of $\sigma$-subalgebras for a vector measure

Ricker, Werner J.

Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 1, pp. 91-97

Résumé

Let $X$ be a locally convex space, $m: \Sigma \to X$ be a vector measure defined on a $\sigma$-algebra $\Sigma$, and $L^1(m)$ be the associated (locally convex) space of $m$-integrable functions. Let $\Sigma(m)$ denote $\{\chi_{{}_{E}}; E\in \Sigma\}$, equipped with the relative topology from $L^1(m)$. For a subalgebra $\Cal A \subseteq \Sigma$, let $\Cal A_\sigma$ denote the generated $\sigma$-algebra and $\overline{\Cal A}_s$ denote the {\sl sequential\/} closure of $\chi(\Cal A) = \{\chi_{{}_{E}}; E\in \Cal A\}$ in $L^1(m)$. Sets of the form $\overline{\Cal A}_s$ arise in criteria determining separability of $L^1(m)$; see [6]. We consider some natural questions concerning $\overline{\Cal A}_s$ and, in particular, its relation to $\chi(\Cal A_\sigma)$. It is shown that $\overline{\Cal A}_s \subseteq \Sigma (m)$ and moreover, that $\{E\in \Sigma; \chi_{{}_{E}} \in \overline{\Cal A}_s\}$ is always a $\sigma$-algebra and contains $\Cal A_\sigma$. Some properties of $X$ are determined which ensure that $\chi(\Cal A_\sigma) = \overline{\Cal A}_s$, for any $X$-valued measure $m$ and subalgebra $\Cal A \subseteq \Sigma$; the class of such spaces $X$ turns out to be quite extensive.

Détail

Classification : 28B05
Keywords: $\sigma $-subalgebra; vector measure; sequential closure