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
Article

Summary

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.

Keywords

Control Theory Recent Result Integral Representation Vector Mapping Unit Sphere 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Berliocchi -J. M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. de France,101 (1973), pp. 129–184.MathSciNetGoogle Scholar
  2. [2]
    J. R. Bismut,Intégrales convexes et probabilités, I.R.I.A. (1972), Domaine de Voluceau, 78-Rocquencourt, France.Google Scholar
  3. [3]
    N. Bourbaki,Topologie générale, Chapitre 9, Hermann, Paris (1958).Google Scholar
  4. [4]
    N. Bourbaki,Espaces vectoriels topologiques, Chapitre 2, Hermann, Paris (1953).Google Scholar
  5. [5]
    N. Bourbaki,Intégration sur les espaces topologiques séparés, Chapitre 9, Hermann, Paris (1969).Google Scholar
  6. [6]
    Ch. Castaing,Un théorème de densité intervenant dans la théorie des « Relaxed controls », Collège Scientifique Universitaire de Perpignan, France (1968).Google Scholar
  7. [7]
    Ch. Castaing - M. Valadier,Convex analysis and measurable multifunctions (to appear).Google Scholar
  8. [8]
    A. Coste,Set-valued measures, Faculté des Sciences de Caen, France (1974).Google Scholar
  9. [9]
    J. P. Daures,Quelques propriétés des espérances conditionnelles des intégrandes convexes de Caratheodory, Séminaire d'Analyse Convexe, Montpellier (1973), Exposé n. 5.Google Scholar
  10. [10]
    G. Debreu - D. Schmeidler,The Radon-Nikodym derivative of a correspondance, Proceedings of the Sixth Berkeley Symposium Math. Stat. and Prob., Berkeley, California (1970).Google Scholar
  11. [11]
    N. Dinculeanu,Vector Measures, Pergamon Press (1967).Google Scholar
  12. [12]
    N. Dunford - J. Schwartz,Linear Operators, Part 1, New York Interscience Publishers (1958).Google Scholar
  13. [13]
    J. Gil de la Madrid,Measures and Tensors, Trans. Amer. Math. Soc.,114 (1965), pp. 98–121.CrossRefMathSciNetGoogle Scholar
  14. [14]
    C. Godet-Thobie,Sur les multimesures de transition, Séminaire d'Analyse Convexe, Montpellier (1974), Exposé n. 5.Google Scholar
  15. [15]
    A. Grothendieck,Produits tensoriels topologiques et espaces nucléaires, Memoirs of the Ann. Math. Soc. (1966).Google Scholar
  16. [16]
    G. W. Johnson, The dual of\(\mathfrak{C}(S,F)\), Math. Ann.,187 (1970), pp. 1–8.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    J. Von Neumann,On rings of operators, Ann. of Math.,50 (1949), pp. 401–485.CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    J. Neveu,Martingales à temps discret, Masson, Paris (1972).Google Scholar
  19. [19]
    C. Olech,The characterization of weak* closure of certain sets of integrable functions, S.I.A.M. Journal on Control,12 (1974), pp. 311–318.MATHMathSciNetGoogle Scholar
  20. [20]
    R. Pallu de la Barriere,Quelques propriétés des multimesures, Séminaire d'Analyse Convexe, Montpellier (1973), Exposé n. 11.Google Scholar
  21. [21]
    A. M. Rous,Problèmes de relaxation pour les équations différentielles à commandes dans les espaces de Banach, Faculté des Sciences de Montpellier, France (1971).Google Scholar
  22. [22]
    M.-F. Sainte-Beuve,On the extension of Von-Neuman-Aumann's theorem, J. Functional Analysis,17 (1974), pp. 112–129.CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    F. Treves,Topological vector spaces, distributions and Kernels, Academic Press (1967).Google Scholar
  24. [24]
    A. Tulcea - C. Tulcea,Topics in the theory of liftings, Springer-Verlag (1969).Google Scholar
  25. [25]
    M. Valadier,Expérance conditionnelle d'un convexe fermé aléatoire, Séminaire d'Analyse Convexe, Montpellier (1972), Exposé n. 1.Google Scholar
  26. [26]
    B. Van Cutsem,Eléments aléatoires à valeurs convexes compactes, Thèse, Grenoble, France (1971).Google Scholar
  27. [27]
    W. Warga,Functions of relaxed controls, S.I.A.M. Journal on Control,5 (1967), pp. 628–641.MATHMathSciNetGoogle Scholar

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

Personalised recommendations