Abstract

If X is a separable Banach space with RNP and Γ is an Effros measurable and Pettis integrable multifunction, then the conditional expectation of Γ with respect to a sub-σ-algebra is well described by the set of conditional expectations of selections of Γ. This is possible due to the Castaing representation of Γ that fails in case of non-separable X. Instead of RNP and separability of X we assume that the multimeasure defined by the Pettis integral of Γ is rich in selections possessing strongly measurable Radon-Nikodym derivatives. In general that cannot be reduced to a separable space. Then, using a lifting, we prove the existence of an Effros measurable conditional expectation of Γ in case of an arbitrary non-separable X and present its representation in terms of quasi-selections of Γ. We apply then the description to martingales of Pettis integrable multifunctions obtaining a scalarly equivalent martingale of measurable multifunctions with many martingale selections.

Author information

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Keywords

  • Multimeasures
  • multifunctions
  • Pettis integral
  • Radon-Nikodym property
  • conditional expectation
  • martingales
  • rich multimeasures

Mathematics Subject Classification

28B20; 28B05, 46G10, 46B22

References

  1. G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer Academic Press, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  2. D. Candeloro, L. Di Piazza, K. Musiał, A. R. Sambucini: Integration of multifunctions with closed convex values in arbitrary Banach spaces, J. Convex Analysis 27 (2020) 1233–1246.https://doi.org/10.68381/jca27065Google Scholar
  3. B. Cascales, V. Kadets, J. Rodriguez: Measurable selectors and set-valued Pettis integral in non-separable Banach spaces, J. Funct. Analysis 256/3 (2009) 673–699.https://doi.org/10.1016/j.jfa.2008.10.022Google Scholar
  4. B. Cascales, V. Kadets, J. Rodriguez: Radon-Nikodym theorems for multimeasures in non-separable spaces, J. Math. Physics, Analysis, Geometry 9 (2013) 7–24.Google Scholar
  5. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  6. S. D. Chatterji: Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968) 21–41.https://doi.org/10.7146/math.scand.a-10868Google Scholar
  7. A. Choukairi-Dini: M-convergence, et régularité des martingales multivoques: Epi-martingales, J. Multivar. Analysis 33 (1990) 39–71.https://doi.org/10.1016/0047-259X(90)90005-3Google Scholar
  8. A. Costé: Set-valued measures, in: Proc. Conf. Topology and Measure I, Zinnowitz 1974, Wiss. Beiträge Ernst-Moritz-Arndt-Universität, Greifswald (1975) 55–74.Google Scholar
  9. A. Costé: La propriété de Radon-Nikodym en intégration multivoque, C. R. Acad. Sci. Paris 280 (1975) 1515–1518.Google Scholar
  10. L. Di Piazza, K. Musiał, A. R. Sambucini: Representations of multimeasures via the multivalued Bartle-Dunford-Schwartz integral, J. Convex Analysis 29 (2022) 1119–1148.https://doi.org/10.68381/jca29062Google Scholar
  11. J. Diestel, J. J. Uhl: Vector Measures, Mathematical Surveys 15, American Mathematical Society, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  12. S. Eder: Some aspects of the lifting property of L∞ (µ, X), Quaestiones Math. 21 (1998) 269–288.https://doi.org/10.1080/16073606.1998.9632046Google Scholar
  13. M. El Allali, M. El-Louh, F. Ezzaki: On convergence and regularity of Aumann-Pettis integrable multivalued martingales, J. Convex. Analysis 27 (2020) 1177–1194.https://doi.org/10.68381/jca27062Google Scholar
  14. K. El Amri, C. Hess: On the Pettis integral of closed valued multifunctions, Set-Valued Analysis 8 (2000) 329–360.https://doi.org/10.1023/A:1026547222209Google Scholar
  15. C. Hess: On multivalued martingales whose values may be unbounded:martingale selectors and Mosco convergence, J. Multivar. Analysis 39 (1991) 175–201.https://doi.org/10.1016/0047-259X(91)90012-QGoogle Scholar
  16. C. Hess: Set-valued integration and set-valued probability theory: an overview, in: Handbook of Measure Theory I, North-Holland, Amsterdam (2002) 617–673.https://doi.org/10.1016/B978-044450263-6/50015-4Google Scholar
  17. S. Hu, N. S. Papageorgiou: Handbook of Multivalued Analysis I, Kluwer Academic Publishers, Dordrecht (1997).https://doi.org/10.1007/978-1-4615-6359-4Google Scholar
  18. A. Ionescu Tulcea, C. Ionescu Tulcea: Topics in the Theory of Lifting, Springer, Berlin (1969).https://doi.org/10.1007/978-3-642-88507-5Google Scholar
  19. D. A. Kandilakis: On the extension of multimeasures and integration with respect to a multimeasure, Proc. Amer. Math. Soc. 116 (1992) 85–92.https://doi.org/10.1090/S0002-9939-1992-1091184-9Google Scholar
  20. K. Musiał: Topics in the theory of Pettis integration, Rend. Ist. Mat. Univ. Trieste XXIII (1991) 177–262.Google Scholar
  21. K. Musiał: Pettis Integral, in: Handbook of Measure Theory I, North-Holland, Amsterdam (2002) 532–586.Google Scholar
  22. K. Musiał: Pettis integrability of multifunctions with values in arbitrary Banach spaces, J. Convex Analysis 18/3 (2011) 769–810.https://doi.org/10.68381/jca18049Google Scholar
  23. K. Musiał: Approximation of Pettis integrable multifunctions with values in arbitrary Banach spaces, J. Convex Analysis 20/3 (2013) 833–870.https://doi.org/10.68381/jca20045Google Scholar
  24. K. Musiał: Multimeasures with values in conjugate Banach spaces and the Weak Radon-Nikodým property, J. Convex Analysis 28/3 (2021) 879–902.https://doi.org/10.68381/jca28051Google Scholar
  25. K. Musiał: Lifting approach to integral representations of rich multimeasures with values in Banach spaces I, J. Convex Analysis 31/1 (2024) 179–193.https://doi.org/10.68381/jca31011Google Scholar
  26. J. Neveu: Bases Mathématiques du Calcul des Probabilités, Masson, Paris (1964).Google Scholar
  27. Y. Sonntag, C. Zalinescu: Scalar convergence of convex sets, J. Math. Analysis Appl. 164 (1992) 219–241.https://doi.org/10.1016/0022-247X(92)90154-6Google Scholar
  28. M. Talagrand: Pettis Integral and Measure Theory, Memoirs of the AMS 307, American Mathematical Society, Providence (1984).https://doi.org/10.1090/memo/0307Google Scholar

Suggested citation

K. Musiał. “Lifting Approach to Integral Representations of Rich Multimeasures with Values in Banach Spaces II.” Journal of Convex Analysis 31 (2024), No. 3, 867–888. https://doi.org/10.68381/jca31043

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.