Ararat, Çağın; Rudloff, Birgit A characterization theorem for Aumann integrals. (English) Zbl 1315.28009 Set-Valued Var. Anal. 23, No. 2, 305-318 (2015). 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. Cited in 3 Documents 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 Keywords:Aumann integral; characterization theorem; Daniell-Stone theorem; closed convex upper sets; complete lattice approach Software:BENSOLVE PDFBibTeX XMLCite \textit{Ç. Ararat} and \textit{B. Rudloff}, Set-Valued Var. Anal. 23, No. 2, 305--318 (2015; Zbl 1315.28009) Full Text: DOI arXiv References: [1] Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1-12 (1965) · Zbl 0163.06301 [2] Çınlar, E.: Probability and Stochastics. Springer, New York (2012) · Zbl 1226.60001 [3] Hamel, A.H.: A duality theory for set-valued functions I: Fenchel conjugation theory. Set-Valued Var. Anal. 17, 153-182 (2009) · Zbl 1168.49031 [4] Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C.: Set optimization—a rather short introduction. In: Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C. (eds.) Set Optimization and Applications in Finance - The State of the Art. Springer, forthcoming in 2014 · Zbl 1337.49001 [5] Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5, 1-28 (2011) · Zbl 1275.91077 [6] Hess, C.: Set-valued integration and set-valued probability theory: an overview. In: Pap, E. (ed.) Handbook of Measure Theory. Elsevier Science B. V. (2002) · Zbl 1022.60011 [7] Kisielewicz, M.: Stochastic Differential Inclusions and Applications. Springer, New York (2013) · Zbl 1277.93002 [8] Löhne, A.: Vector Optimization with Infimum and Supremum. Springer, Berlin Heidelberg (2011) · Zbl 1230.90002 [9] Molchanov, I.: Theory of Random Sets. Springer, London (2005) · Zbl 1109.60001 [10] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401 [11] Rockafellar, R. T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin Heidelberg (1998) , third printing 2009 · Zbl 0888.49001 [12] Stone, M.H.: Notes on integration, II. Proc. Natl. Acad. Sci. USA 34, 447-455 (1948) · Zbl 0034.02903 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.