Functiones et Approximatio Commentarii Mathematici

Mackey topologies and compactness in spaces of vector measures

Marian Nowak

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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

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

First available in Project Euclid: 27 March 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

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


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.

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  • C.D. Aliprantis, and O. Burkinshaw, Positive Operators, Academic Press, New York, 1985.
  • N. Bourbaki, Elements of Mathematics, Topological Vector Spaces, Springer-Verlag, Berlin, 1987, Chap. 1–5.
  • J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math., vol. 92, Springer-Verlag, 1984.
  • J. Diestel and J.J. Uhl, Vector Measures, Amer. Math. Soc., Math. Surveys 15, Providence, RI, 1977.
  • W.H. Graves, On the theory of vector measures, Amer. Math. Soc. Memoirs 195 (1977), Providence, R.I.
  • W.H. Graves and W. Ruess, Compactness in spaces of vector-valued measures and a natural Mackey topology for spaces of bounded measurable functions, Contemp. Math. 2 (1980), 189–203.
  • A. Grothendieck, Sur les applications lineaires faiblement compactes d'espaces de type $C(K)$, Canad. J. Math. 5 (1953), 129–173.
  • S.S. Khurana, A topology associated with vector measures, J. Indian Math. Soc. 45 (1981), 167–179.
  • D.R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), no. 1, 157–165.
  • M. Nowak, Vector measures and Mackey topologies, Indag. Math. 23 (2012), 113–122.
  • T.V. Panchapagesan, Applications of a theorem of Grothendieck to vector measures, J. Math. Anal. Appl. 214 (1997), 89–101.
  • T.V. Panchapagesan, Characterizations of weakly compact operators on $C_0(T)$, Trans. Amer. Math. Soc. 350 (1998), no. 12, 4849–4867.
  • H. Schaefer and Xiao-Dong Zhang, On the Vitali-Hahn-Saks Theorem, in: Operator Theory: Advances and Applications, vol. 75, Birkhäuser, Basel, 1995, pp. 289–297.
  • Xiao-Dong Zhang, On weak compactness is spaces of measures, J. Func. Anal. 143 (1997), 1–9.