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Annali di Matematica Pura ed Applicata

, Volume 101, Issue 1, pp 229–236 | Cite as

Multifunctions on abstract measurable spaces and application to stochastic decision theory

  • C. J. Himmelberg
  • F. S. Van Vleck
Article

Summary

The main results are some very general theorems about measurable multifunctions on abstract measurable spaces with compact values in a separable metric space. It is shown that measurability is equivalent to the existence of a pointwise dense countable family of measurable selectors, and that the intersection of two compact-valued measurable multifunctions is measurable. These results are used to obtain a Filippov type implicit function theorem, and a general theorem concerning the measurability of y(t)=min f({t} × Γ(t)) when f is a real valued function and Γ a compact valued multifunction. An application to stochastic decision theory is given generalizing a result of Benes.

Keywords

Measurable Space Decision Theory Implicit Function Theorem General Theorem Countable Family 
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.

References

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    V. E. Benes,Existence of optimal strategies based on specified information, for a class of stochastic decision problems, SIAM J. Control,8 (1970), pp. 179–188.CrossRefzbMATHMathSciNetGoogle Scholar
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    C. Castaing,Sur les multi-applications mesurables, Revue Française d'Informatique et de Recherche Operationelle,1 (1967), pp. 91–126.zbMATHMathSciNetGoogle Scholar
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    S. T. Hu,Theory of Retracts, Wayne State University Press, Detroit, 1965.Google Scholar
  4. [HV]
    C. J. Himmelberg -F. S. Van Vleck,Some selection theorems for measurable functions, Canad. J. Math.,21 (1969), pp. 394–399.MathSciNetGoogle Scholar
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    K. Kuratowski,Topology, vol. I, Academic Press, New York, 1966.Google Scholar
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    R. T. Rockafeller,Measurable dependence of convex sets and functions on parameters, J. Math. Anal. Appl.,28 (1969), pp. 4–25.CrossRefMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1974

Authors and Affiliations

  • C. J. Himmelberg
    • 1
  • F. S. Van Vleck
    • 1
  1. 1.Lawrence

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