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Journal of Convex Analysis 18 (2011), No. 3, 873--895
Copyright Heldermann Verlag 2011

The Gelfand Integral for Multi-Valued Functions

Bernardo Cascales
Dep. de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain

Vladimir Kadets
Dept. of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine

José Rodríguez
Dep. de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100 Espinardo-Murcia, Spain

We study the existence of w*-scalarly measurable selectors and almost selectors for w*-scalarly measurable multi-functions with values in dual Banach spaces. These selection results are used to study Gelfand and Dunford integrals for multi-functions: our non separable setting extends previous studies that have been done for separable Banach spaces. Pettis integral for multi-functions, already studied by different authors, naturally appears as a particular case of Dunford integral. We also study when the Gelfand integral of a multi-function is not only w*-compact but w-compact.

Keywords: Multi-function, measurable selector, Gelfand integral for multi-functions, Dunford integral for multi-functions, Pettis integral for multi-functions.

MSC: 28B05, 28B20, 46G10

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