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A Characterization Theorem for Aumann Integrals

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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.

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Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, 08544, USA

    Çağın Ararat

  2. Department of Operations Research and Financial Engineering & Bendheim Center for Finance, Princeton University, Princeton, NJ, 08544, USA

    Birgit Rudloff

Authors
  1. Çağın Ararat
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  2. Birgit Rudloff
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Correspondence to Birgit Rudloff.

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Ararat, Ç., Rudloff, B. A Characterization Theorem for Aumann Integrals. Set-Valued Var. Anal 23, 305–318 (2015). https://doi.org/10.1007/s11228-014-0309-0

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  • DOI: https://doi.org/10.1007/s11228-014-0309-0

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