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A Characterization Theorem for Aumann Integrals

Author(s): Ararat, Çağın; Rudloff, Birgit

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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.
Publication Date: 14-Nov-2014
Citation: Ararat, Çağın, and Birgit Rudloff. "A characterization theorem for Aumann integrals." Set-Valued and Variational Analysis 23, no. 2 (2015): 305-318. doi:10.1007/s11228-014-0309-0
DOI: 10.1007/s11228-014-0309-0
ISSN: 1877-0533
EISSN: 1877-0541
Pages: 305 - 318
Type of Material: Journal Article
Journal/Proceeding Title: Set-Valued and Variational Analysis
Version: Author's manuscript

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