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

Ricker, Werner J.. "Sequential closures of $\sigma $-subalgebras for a vector measure." Commentationes Mathematicae Universitatis Carolinae 37.1 (1996): 91-97. <http://eudml.org/doc/247896>.

@article{Ricker1996,
abstract = {Let $X$ be a locally convex space, $m: \Sigma \rightarrow X$ 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 $\lbrace \chi _\{\{\}_\{E\}\}; E\in \Sigma \rbrace $, equipped with the relative topology from $L^1(m)$. For a subalgebra $\mathcal \{A\} \subseteq \Sigma $, let $\mathcal \{A\}_\sigma $ denote the generated $\sigma $-algebra and $\overline\{\mathcal \{A\}\}_s$ denote the sequential closure of $\chi (\mathcal \{A\}) = \lbrace \chi _\{\{\}_\{E\}\}; E\in \mathcal \{A\}\rbrace $ in $L^1(m)$. Sets of the form $\overline\{\mathcal \{A\}\}_s$ arise in criteria determining separability of $L^1(m)$; see [6]. We consider some natural questions concerning $\overline\{\mathcal \{A\}\}_s$ and, in particular, its relation to $\chi (\mathcal \{A\}_\sigma )$. It is shown that $\overline\{\mathcal \{A\}\}_s \subseteq \Sigma (m)$ and moreover, that $\lbrace E\in \Sigma ; \chi _\{\{\}_\{E\}\} \in \overline\{\mathcal \{A\}\}_s\rbrace $ is always a $\sigma $-algebra and contains $\mathcal \{A\}_\sigma $. Some properties of $X$ are determined which ensure that $\chi (\mathcal \{A\}_\sigma ) = \overline\{\mathcal \{A\}\}_s$, for any $X$-valued measure $m$ and subalgebra $\mathcal \{A\} \subseteq \Sigma $; the class of such spaces $X$ turns out to be quite extensive.},
author = {Ricker, Werner J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\sigma $-subalgebra; vector measure; sequential closure; -subalgebra; sequential closure; vector measure},
language = {eng},
number = {1},
pages = {91-97},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Sequential closures of $\sigma $-subalgebras for a vector measure},
url = {http://eudml.org/doc/247896},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Ricker, Werner J.
TI - Sequential closures of $\sigma $-subalgebras for a vector measure
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 1
SP - 91
EP - 97
AB - Let $X$ be a locally convex space, $m: \Sigma \rightarrow X$ 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 $\lbrace \chi _{{}_{E}}; E\in \Sigma \rbrace $, equipped with the relative topology from $L^1(m)$. For a subalgebra $\mathcal {A} \subseteq \Sigma $, let $\mathcal {A}_\sigma $ denote the generated $\sigma $-algebra and $\overline{\mathcal {A}}_s$ denote the sequential closure of $\chi (\mathcal {A}) = \lbrace \chi _{{}_{E}}; E\in \mathcal {A}\rbrace $ in $L^1(m)$. Sets of the form $\overline{\mathcal {A}}_s$ arise in criteria determining separability of $L^1(m)$; see [6]. We consider some natural questions concerning $\overline{\mathcal {A}}_s$ and, in particular, its relation to $\chi (\mathcal {A}_\sigma )$. It is shown that $\overline{\mathcal {A}}_s \subseteq \Sigma (m)$ and moreover, that $\lbrace E\in \Sigma ; \chi _{{}_{E}} \in \overline{\mathcal {A}}_s\rbrace $ is always a $\sigma $-algebra and contains $\mathcal {A}_\sigma $. Some properties of $X$ are determined which ensure that $\chi (\mathcal {A}_\sigma ) = \overline{\mathcal {A}}_s$, for any $X$-valued measure $m$ and subalgebra $\mathcal {A} \subseteq \Sigma $; the class of such spaces $X$ turns out to be quite extensive.
LA - eng
KW - $\sigma $-subalgebra; vector measure; sequential closure; -subalgebra; sequential closure; vector measure
UR - http://eudml.org/doc/247896
ER -