Sequential closures of $\sigma$-subalgebras for a vector measure
Ricker, Werner J.
Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996), p. 91-97 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 1996-01-01
Classification:  28B05 
@article{118814,
     author = {Werner J. Ricker},
     title = {Sequential closures of $\sigma$-subalgebras for a vector measure},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {37},
     year = {1996},
     pages = {91-97},
     zbl = {0877.28011},
     mrnumber = {1396162},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118814}
}

Ricker, Werner J. Sequential closures of $\sigma$-subalgebras for a vector measure. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 91-97. http://gdmltest.u-ga.fr/item/118814/

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