Abstract

It has been proven by Cascales, Kadets and Rodriguez [J. Convex Anal. 18 (2011), 873-895] that each weak∗^* scalarly integrable multifunction (with respect to a probability measure μ, whose values are compact convex subsets of a conjugate Banach space X∗X^* and the family of support functions determined by X is order bounded in L1(μ)L_1(\mu), is Gelfand integrable in the family of weakly compact convex subsets of X∗X^*. A question has been posed whether a similar result holds true for multifunctions with weakly compact convex values. We prove that the answer is affirmative if X does not contain any isomorphic copy of l1l_1. If moreover the multifunction is compact valued, then it is Gelfand integrable in the family of compact convex subsets of X∗X^*.

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Kazimierz Musiał

Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

musial@math.uni.wroc.pl

Keywords

  • Multifunction
  • Gelfand set-valued integral
  • Pettis set-valued integral
  • support function

Mathematics Subject Classification

28B20; 28B05, 46G10, 54C60

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Suggested citation

K. Musiał. “Gelfand Integral of Multifunctions.” Journal of Convex Analysis 21 (2014), No. 4, 1193–1200. https://doi.org/10.68381/jca21062

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.