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A characterization theorem for Aumann integrals. (English) Zbl 1315.28009

Summary: 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.

MSC:

28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
26E25 Set-valued functions
46B42 Banach lattices
54C60 Set-valued maps in general topology

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References:

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