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Remarks on the range of a vector measure

Published online by Cambridge University Press:  18 May 2009

Jesús M. F. Castilo  and
Fernando Sánchez
Jesús M. F. Castilo
Affiliation:
Departamento de Matematicas, Universidad de Extremadura, Avda. de Elvas, S/N. 06071 Badajoz, España.
Fernando Sánchez
Affiliation:
Departamento de Matematicas, Universidad de Extremadura, Avda. de Elvas, S/N. 06071 Badajoz, España.
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A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 2 , May 1994 , pp. 157 - 161
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

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