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Extension of a Bounded Vector Measure with Values in a Reflexive Banach Space

Published online by Cambridge University Press:  20 November 2018

Geoffrey Fox
Geoffrey Fox*
Affiliation:
Université de Montréal
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A vector measure (countable additive set function with values in a Banach space) on a field may be extended to a vector measure on the generated σ- field, under certain hypotheses. For example, the extension is established for the bounded variation case [2, 5, 8], and there are more general conditions under which the extension exists [ 1 ]. The above results have as hypotheses fairly strong boundedness conditions on the n o rm of the measure to be extended. In this paper we prove an extension theorem of the same type with a restriction on the range, supposing further that the measure is merely bounded. In fact a vector measure on a σ- field is bounded (III. 4. 5 of [3]) but it is conceivable that a vector measure on a field could be unbounded.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 10 , Issue 4 , October 1967 , pp. 525 - 529
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Arsene, Gr., Strǎtilǎ, S., Prolongement des mesures vectorielles. Rev. Roumaine Math. Pure. appl. 10 (1966), 333-338.Google Scholar
2. Dinculeanu, N., On Regular Vector Measures. Acta Sci. Math. (Szeged) 24 (1966), 236-243.Google Scholar
3. Dunford-Schwartz, , Linear Operators (part I). Interscience Publishers, New York (1955).Google Scholar
4. Fox, G., Note on the Borel method of measure extension. Canad. Math. Bull. 5 (1966), 285-296.Google Scholar
5. Gǎinǎ, S., Extension of vector measures with finite variation. Rev. Roumaine Math. Pure. appl. 8 (1966), 151-154.Google Scholar
6. Halmos, P. R., Measure Theory. D. Van Nostrand (1955).Google Scholar
7. Le Blanc, L. and Fox, G. E., On the extension of measure by the method of Borel. Canad. Jour. Math. 8 (1955), 516-523.Google Scholar
8. Nicolescu, M., Mathematical Analysis. (Romanian. Vol. Ill Ed. Technica, Bucharest (1960).Google Scholar