Functiones et Approximatio Commentarii Mathematici

Mackey topologies and compactness in spaces of vector measures

Marian Nowak

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Abstract

Let $\Sigma$ be a $\sigma$-algebra of subsets of a non-empty set $\Omega$. Let $B(\Sigma)$ be the space of all bounded $\Sigma$-measurable scalar functions defined on $\Omega$, equipped with the natural Mackey topology $\tau(B(\Sigma),ca(\Sigma))$. Let $(E,\xi)$ be a quasicomplete locally convex Hausdorff space and let $ca(\Sigma,E)$ be the space of all $\xi$-countably additive $E$-valued measures on $\Sigma$, provided with the topology ${\cal T}_s$ of simple convergence. We characterize relative ${\cal T}_s$-compactness in $ca(\Sigma,E)$, in terms of the topological properties of the corresponding sets in the space ${\cal L}_{\tau,\xi}(B(\Sigma),E)$ of all $(\tau(B(\Sigma),ca(\Sigma)),\xi)$-continuous integration operators from $B(\Sigma)$ to $E$. A generalized Nikodym type convergence theorem is derived.

Article information

Source
Funct. Approx. Comment. Math., Volume 50, Number 1 (2014), 191-198.

Dates
First available in Project Euclid: 27 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1395924292

Digital Object Identifier
doi:10.7169/facm/2014.50.1.8

Mathematical Reviews number (MathSciNet)
MR3189508

Zentralblatt MATH identifier
1298.46025

Subjects
Primary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]
Secondary: 28A25: Integration with respect to measures and other set functions 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx] 47B38: Operators on function spaces (general)

Keywords
spaces of bounded measurable functions Mackey topologies strongly Mackey space vector measures integration operators topology of simple convergence

Citation

Nowak, Marian. Mackey topologies and compactness in spaces of vector measures. Funct. Approx. Comment. Math. 50 (2014), no. 1, 191--198. doi:10.7169/facm/2014.50.1.8. https://projecteuclid.org/euclid.facm/1395924292


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