Journal of Convex Analysis 18 (2011), No. 3, 873--895
Copyright Heldermann Verlag 2011
The Gelfand Integral for Multi-Valued Functions
Dep. de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain
Dept. of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine
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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