A Characterization of Weak Radon-Nikodym Sets In Dual Banach Spaces

Throughout this paper X and Y denote real Banach spaces with topological duals X* and Y* respectively. The closed unit ball in X is denoted by Bx* In the following, (fi, S, //) always denotes a complete finite measure space and ([0, 1], A, X) is the Lebesgue measure space on [0, !]„ For each (Q, S3 //), a function f : Q->X is said to be weakly measurable if for each x*^X* the real-valued function (#*5 f(co)) is /^-measurable. We say that a weakly measurable function/: Q-+X is Pettis integrable if (**, /(o>)) e I* (12, I, /*) for every x*£=iX* and moreover for each E^S there exists an element xE^X that satisfies