## Functiones et Approximatio Commentarii Mathematici

### Mackey topologies and compactness in spaces of vector measures

Marian Nowak

#### 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

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

#### 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

#### References

• 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.