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Set-Valued and Variational Analysis

, Volume 23, Issue 2, pp 305–318 | Cite as

A Characterization Theorem for Aumann Integrals

  • Çağın Ararat
  • Birgit RudloffEmail author
Article

Abstract

A Daniell-Stone type characterization theorem for Aumann integrals of set-valued measurable functions will be proven. It is assumed that the values of these functions are closed convex upper sets, a structure that has been used in some recent developments in set-valued variational analysis and set optimization. It is shown that the Aumann integral of such a function is also a closed convex upper set. The main theorem characterizes the conditions under which a functional that maps from a certain collection of measurable set-valued functions into the set of all closed convex upper sets can be written as the Aumann integral with respect to some σ-finite measure. These conditions include the analog of the conlinearity and monotone convergence properties of the classical Daniell-Stone theorem for the Lebesgue integral, and three geometric properties that are peculiar to the set-valued case as they are redundant in the one-dimensional setting.

Keywords

Aumann integral Characterization theorem Daniell-Stone theorem Closed convex upper sets Complete lattice approach 

Mathematics Subject Classifications (2010)

28B20 26E25 46B42 54C60 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA
  2. 2.Department of Operations Research and Financial Engineering & Bendheim Center for FinancePrinceton UniversityPrincetonUSA

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