W.J. Ricker
Sequential closures of $\sigma $-subalgebras for a vector measure

Comment.Math.Univ.Carolinae 37,1 (1996) 89-95.

Abstract: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 {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.

Keywords: $\sigma $-subalgebra, vector measure, sequential closure
AMS Subject Classification: 28B05