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Vector measures and nuclearity. (English) Zbl 0654.46051
Let X be a locally convex space topologized by seminorms \(\{p_{\lambda}\}\). Using X valued measures on \(\delta\)-rings (equiv. semi-tribe or \(\delta\)-clan, a concept containing \(\sigma\)-ring of measurable sets), the author obtained some characterizations of nuclearity of X. The \(\lambda\)-variation is defined as \(m_{\lambda}(E)\equiv \sup \sum p_{\lambda}(m(E_ i))\), where sup ranges over finite coverings of E. A measure m is called of finite variation if \(m_{\lambda}<\infty\) for any \(\lambda\). The results are
1. Suppose X metrizable or dually-metrizable. If any X-valued measure has finite variation then X is nuclear.
2. Moreover if X is completely metrizable or completely dually-metrizable then X is reflexive.
3. Conversely any dual-nuclear space valued measure has finite variation.
Reviewer: S.Takenaka

MSC:
46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
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References:
[1] DINCULEANU N.: Vector Measures. Berlin 1966. · Zbl 0142.10502
[2] PIETSCH A.: Nukleare lokalkonvexe Raume. Berlin 1965. · Zbl 0152.32302
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