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Journal of Convex Analysis 21 (2014), No. 4, 1193--1200
Copyright Heldermann Verlag 2014

Gelfand Integral of Multifunctions

Kazimierz Musial
Institute of Mathematics, Wroclaw University, Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

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* and the family of support functions determined by X is order bounded in L1(μ), is Gelfand integrable in the family of weakly compact convex subsets of 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 l1. If moreover the multifunction is compact valued, then it is Gelfand integrable in the family of compact convex subsets of X*.

Keywords: Multifunction, Gelfand set-valued integral, Pettis set-valued integral, support function.

MSC: 28B20; 28B05, 46G10, 54C60

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