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arXiv · 1404.7440

A Characterization Theorem for Aumann Integrals

Çağın Ararat·Birgit Rudloff
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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 $\sigma$-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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Çağın Ararat, Birgit Rudloff. 2014-04-28. A Characterization Theorem for Aumann Integrals. https://doi.org/10.1007/s11228-014-0309-0

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