A Generalization of the Radon–Nikodym Property in Dual Banach Spaces, Fragmentedness, and Differentiability of Convex Functions

Abstract

For non-empty bounded subsets A of Banach spaces, we introduce the notion of the A-Radon–Nikodym property in dual Banach spaces, a slight generalization of the Radon–Nikodym property in such spaces. Making the effective use of this notion and a weak*-measurable function constructed here, we give a direct study of some related properties (especially, A-fragmentedness) of weak*-compact subsets of dual Banach spaces.