Criteria for closedness of vector measures

Author:
W. Ricker

Journal:
Proc. Amer. Math. Soc. **91** (1984), 75-80

MSC:
Primary 28B05; Secondary 46G10

MathSciNet review:
735568

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that there is a large class of locally convex spaces, which includes, for example, all spaces which are metrizable or Suslin or have the strict Mackey convergence property as well as many dual spaces, with the property that any vector measure assuming its values in such a space is a closed measure.

**[1]**Nicolas Bourbaki,*Éléments d’histoire des mathématiques*, Hermann, Paris, 1974 (French). Nouvelle édition revue, corrigée et augmentée; Histoire de la Pensée, No. IV. MR**0349314****[2]**Cecilia H. Brook and William H. Graves,*Closed measures*, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 145–160. MR**621856****[3]**Klaus Floret,*Weakly compact sets*, Lecture Notes in Mathematics, vol. 801, Springer, Berlin, 1980. Lectures held at S.U.N.Y., Buffalo, in Spring 1978. MR**576235****[4]**Igor Kluvánek,*The range of a vector-valued measure*, Math. Systems Theory**7**(1973), 44–54. MR**0322131****[5]**Igor Kluvánek,*Conical measures and vector measures*, Ann. Inst. Fourier (Grenoble)**27**(1977), no. 1, v, 83–105 (English, with French summary). MR**0470173****[6]**I. Kluvánek and G. Knowles,*Vector measures and control systems*, North-Holland, Amsterdam, 1976.**[7]**G. Köthe,*Topological vector spaces*. I, Die Grundlehren der Math. Wissenschaften, No. 159, Springer-Verlag, Berlin, 1969.**[8]**W. Ricker,*On Boolean algebras of projections and scalar-type spectral operators*, Proc. Amer. Math. Soc.**87**(1983), no. 1, 73–77. MR**677235**, 10.1090/S0002-9939-1983-0677235-6**[9]**Elias Saab,*On the Radon-Nikodým property in a class of locally convex spaces*, Pacific J. Math.**75**(1978), no. 1, 281–291. MR**500045****[10]**G. Erik F. Thomas,*Integration of functions with values in locally convex Suslin spaces*, Trans. Amer. Math. Soc.**212**(1975), 61–81. MR**0385067**, 10.1090/S0002-9947-1975-0385067-1**[11]**François Trèves,*Topological vector spaces, distributions and kernels*, Academic Press, New York-London, 1967. MR**0225131**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
28B05,
46G10

Retrieve articles in all journals with MSC: 28B05, 46G10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0735568-X

Keywords:
Vector measure,
closed measure,
equicontinuous operator-valued measure

Article copyright:
© Copyright 1984
American Mathematical Society