Vector measures and nuclearity. (English) Zbl 0654.46051

Duchoň, Miloslav

Math. Slovaca 38, No. 1, 79-83 (1988).

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.)

Keywords:

characterizations of nuclearity; dual-nuclear space valued measure

References:

[1]	DINCULEANU N.: Vector Measures. Berlin 1966. · Zbl 0142.10502
[2]	PIETSCH A.: Nukleare lokalkonvexe Raume. Berlin 1965. · Zbl 0152.32302