# Article

Full entry | PDF   (0.2 MB)
Keywords:
$\sigma$-subalgebra; vector measure; sequential closure
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.
References:
[1] Bourbaki N.: Topologie générale. II (Nouvelle Édition). Chapitres 5 à 10, Herman, Paris, 1974.
[2] Dunford N., Schwartz J.T.: Linear operators III; spectral operators. Wiley-Interscience, New York, 1972. MR 1009164
[3] Floret K.: Weakly compact sets. Lecture Notes in Math., Vol.801, Springer-Verlag, Berlin and New York, 1980. MR 0576235 | Zbl 0437.46006
[4] Kluvánek I., Knowles G.: Vector measures and control systems. North Holland, Amsterdam, 1976. MR 0499068
[5] Ricker W.J.: Criteria for closedness of vector measures. Proc. Amer. Math. Soc. 91 (1984), 75-80. MR 0735568 | Zbl 0544.28005
[6] Ricker W.J.: Separability of the $L^1$-space of a vector measure. Glasgow Math. J. 34 (1992), 1-9. MR 1145625
[7] Schwartz L.: Radon measures on arbitrary topological spaces and cylindrical measures. Oxford University Press, Bombay, 1973. MR 0426084 | Zbl 0298.28001
[8] Thomas G.E.F.: Integration of functions in locally convex Suslin spaces. Trans. Amer. Math. Soc. 212 (1975), 61-81. MR 0385067

Partner of