Annali di Matematica Pura ed Applicata

, Volume 116, Issue 1, pp 317–379 | Cite as

Some topological properties of vector measures with bounded variation and its applications

  • M. -F. Sainte-Beuve


In this paper, we study the duality between ℜ(T, E′) and\(\mathfrak{C}^b (T) \otimes E\), where ℜ(T, E′) is the space of vector measures with bounded variation, defined on a completely regular Suslin space T, with values in the weak dual space E′ of a separable Fréchet space E;\(\mathfrak{C}^b (T) \otimes E\) is the space of continuous bounded mappings defined on T, with values in a finite dimensional subspace of E. We show that ℜ(T, E′) is a Suslin space when equipped with the weak topology\(\sigma (\mathfrak{M}(T,E'),\mathfrak{C}^b (T) \otimes E)\). Some applications are given: a weak closure's theorem for integrable vector mappings (a generalization of a recent result of Olech), a theorem of integral representation, a theorem of existence of conditional expectation, and finally a theorem of density related to the optimal control theory. A new application of these results is the study of comparison of measures on the unit sphere Sn ofRn, which will be published elsewhere.


Control Theory Recent Result Integral Representation Vector Mapping Unit Sphere 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1978

Authors and Affiliations

  • M. -F. Sainte-Beuve
    • 1
  1. 1.U.S.T.L. MontpellierFrance

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