The Gelfand Integral for Multi-Valued Functions

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.