Abstract

We study the existence of w∗w^*-scalarly measurable selectors and almost selectors for w∗w^*-scalarly measurable multi-functions with values in dual Banach spaces. These selection results are used to study Gelfand and Dunford integrals for multi-functions: our non separable setting extends previous studies that have been done for separable Banach spaces. Pettis integral for multi-functions, already studied by different authors, naturally appears as a particular case of Dunford integral. We also study when the Gelfand integral of a multi-function is not only w∗w^*-compact but w-compact.

Author information

Contact details are reproduced from the original publication and may be historical.

Bernardo Cascales

Dep. de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain

beca@um.es

Vladimir Kadets

Dept. of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine

vova1kadets@yahoo.com

José Rodríguez

Dep. de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100 Espinardo-Murcia, Spain

joserr@um.es

Keywords

  • Multi-function
  • measurable selector
  • Gelfand integral for multi-functions
  • Dunford integral for multi-functions
  • Pettis integral for multi-functions

Mathematics Subject Classification

28B05, 28B20, 46G10

References

  1. F. Albiac, N. J. Kalton: Topics in Banach Space Theory, Graduate Texts in Mathematics 233, Springer, Berlin (2006).Google Scholar
  2. C. D. Aliprantis, K. C. Border: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd Ed., Springer, Berlin (2006).Google Scholar
  3. R. D. Bourgin: Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lecture Notes in Mathematics 993, Springer, Berlin (1983).https://doi.org/10.1007/BFb0069321Google Scholar
  4. B. Cascales, V. Kadets, J. Rodríguez: The Pettis integral for multi-valued functions via single-valued ones, J. Math. Anal. Appl. 332(1) (2007) 1–10.https://doi.org/10.1016/j.jmaa.2006.10.003Google Scholar
  5. B. Cascales, V. Kadets, J. Rodríguez: Measurable selectors and set-valued Pettis integral in non-separable Banach spaces, J. Funct. Anal. 256(3) (2009) 673–699.https://doi.org/10.1016/j.jfa.2008.10.022Google Scholar
  6. B. Cascales, V. Kadets, J. Rodríguez: Measurability and selections of multi-functions in Banach spaces, J. Convex Analysis 17(1) (2010) 229–240.https://doi.org/10.68381/jca17017Google Scholar
  7. B. Cascales, J. Rodríguez: Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297(2) (2004) 540–560.https://doi.org/10.1016/j.jmaa.2004.03.026Google Scholar
  8. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  9. L. Di Piazza, K. Musiał: Set-valued Kurzweil-Henstock-Pettis integral, Set-Valued Anal. 13(2) (2005) 167–179.https://doi.org/10.1007/s11228-004-0934-0Google Scholar
  10. L. Di Piazza, K. Musiał: A decomposition theorem for compact-valued Henstock integral, Monatsh. Math. 148(2) (2006) 119–126.https://doi.org/10.1007/s00605-005-0376-2Google Scholar
  11. L. Di Piazza, K. Musiał: A decomposition of Henstock-Kurzweil-Pettis integrable multifunctions, in: Vector Measures, Integration and Related Topics, G. P. Curbera et al. (ed.), Operator Theory: Advances and Applications 201, Birkhäuser, Basel (2010) 171–182.https://doi.org/10.1007/978-3-0346-0211-2_16Google Scholar
  12. J. Diestel: Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92, Springer, New York (1984).https://doi.org/10.1007/978-1-4612-5200-9Google Scholar
  13. J. Diestel, J. J. Uhl, Jr.: Vector Measures, Mathematical Surveys 15, American Mathematical Society, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  14. M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, V. Zizler: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 8, Springer, New York (2001).https://doi.org/10.1007/978-1-4757-3480-5Google Scholar
  15. I. M. Gel’fand: Sur un lemme de la théorie des espaces linéaires, Commun. Inst. Sci. Math. et Mecan., Univ. Kharkoff et Soc. Math. Kharkoff, IV. Ser. 13 (1936) 35–40.Google Scholar
  16. M. Ali Khan: On the integration of set-valued mappings in a nonreflexive Banach space. II, Simon Stevin 59(3) (1985) 257–267.Google Scholar
  17. M. Ali Khan: Equilibrium points of nonatomic games over a Banach space, Trans. Amer. Math. Soc. 293(2) (1986) 737–749.https://doi.org/10.1090/S0002-9947-1986-0816322-3Google Scholar
  18. M. Ali Khan, Y. Sun: Integrals of set-valued functions with a countable range, Math. Oper. Res. 21(4) (1996) 946–954.https://doi.org/10.1287/moor.21.4.946Google Scholar
  19. M. Ali Khan, Y. Sun: Nonatomic games on Loeb spaces, Proc. Natl. Acad. Sci. USA 93(26) (1996) 15518–15521 (electronic).https://doi.org/10.1073/pnas.93.26.15518Google Scholar
  20. E. Klein, A. C. Thompson: Theory of Correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York (1984).Google Scholar
  21. G. Köthe: Topological Vector Spaces. I, Grundlehren der mathematischen Wissenschaften 159, Springer, New York (1969).Google Scholar
  22. P. A. Loeb, Y. Sun: A general Fatou lemma, Adv. Math. 213(2) (2007) 741–762.https://doi.org/10.1016/j.aim.2007.01.008Google Scholar
  23. K. Musiał: Topics in the theory of Pettis integration, Rend. Ist. Mat. Univ. Trieste 23(1) (1991) 177–262.Google Scholar
  24. K. Podczeck: On the convexity and compactness of the integral of a Banach space valued correspondence, J. Math. Econ. 44(7–8) (2008) 836–852.https://doi.org/10.1016/j.jmateco.2007.03.003Google Scholar
  25. Y. Sun: Integration of correspondences on Loeb spaces, Trans. Amer. Math. Soc. 349(1) (1997) 129–153.https://doi.org/10.1090/S0002-9947-97-01825-4Google Scholar
  26. Y. Sun, N. C. Yannelis: Saturation and the integration of Banach valued correspondences, J. Math. Econ. 44(7–8) (2008) 861–865.https://doi.org/10.1016/j.jmateco.2007.07.005Google Scholar
  27. M. Talagrand: Pettis integral and measure theory, Mem. Amer. Math. Soc. 51(307) (1984) 224p.https://doi.org/10.1090/memo/0307Google Scholar
  28. M. Valadier: Multi-applications mesurables à valeurs convexes compactes, J. Math. Pures Appl., IX. Sér. 50 (1971) 265–297.Google Scholar
  29. D. van Dulst: Characterizations of Banach Spaces not Containing l1, CWI Tracts 59, Centrum voor Wiskunde en Informatica, Amsterdam (1989).Google Scholar
  30. H. Ziat: Convergence theorems for Pettis integrable multifunctions, Bull. Pol. Acad. Sci., Math. 45(2) (1997) 123–137.Google Scholar
  31. H. Ziat: On a characterization of Pettis integrable multifunctions, Bull. Pol. Acad. Sci., Math. 48(3) (2000) 227–230.Google Scholar

Suggested citation

B. Cascales, V. Kadets, J. Rodríguez. “The Gelfand Integral for Multi-Valued Functions.” Journal of Convex Analysis 18 (2011), No. 3, 873–895. https://doi.org/10.68381/jca18055

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.