| Title:
|
Sequential closures of $\sigma$-subalgebras for a vector measure (English) |
| Author:
|
Ricker, Werner J. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
1 |
| Year:
|
1996 |
| Pages:
|
91-97 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
$\sigma $-subalgebra |
| Keyword:
|
vector measure |
| Keyword:
|
sequential closure |
| MSC:
|
28B05 |
| idZBL:
|
Zbl 0877.28011 |
| idMR:
|
MR1396162 |
| . |
| Date available:
|
2009-01-08T18:22:14Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118814 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[4] Kluvánek I., Knowles G.: Vector measures and control systems.North Holland, Amsterdam, 1976. MR 0499068 |
| Reference:
|
[5] Ricker W.J.: Criteria for closedness of vector measures.Proc. Amer. Math. Soc. 91 (1984), 75-80. Zbl 0544.28005, MR 0735568 |
| Reference:
|
[6] Ricker W.J.: Separability of the $L^1$-space of a vector measure.Glasgow Math. J. 34 (1992), 1-9. MR 1145625 |
| Reference:
|
[7] Schwartz L.: Radon measures on arbitrary topological spaces and cylindrical measures.Oxford University Press, Bombay, 1973. Zbl 0298.28001, MR 0426084 |
| Reference:
|
[8] Thomas G.E.F.: Integration of functions in locally convex Suslin spaces.Trans. Amer. Math. Soc. 212 (1975), 61-81. MR 0385067 |
| . |
