The Geometry of Weak Radon-Nikodym Sets in Dual Banach Spaces

Pettis Decomposition for Universally Scalarly Measurable Functions

“…Since / is universally scalarly measurable, it is easy to see that fy is universally scalarly measurable also. Hence by Theorem 6 of [8], fy is Pettis integrable. However, the proof of this theorem yields the fact that {yfy : y G By} is almost weakly precompact in 7/°°(/¿).…”

“…In [8], Riddle, Saab, and Uhl show that if A is a separable Banach space, (fi,E,/t) a finite measure space, and /: fi -> X* a bounded and weak* scalarly measurable function such that {xf: x G Bx} is almost weakly precompact in L°°(/¿), then / is /¿-Pettis integrable.…”

Representations of dynamical systems on Banach spaces not containing $l_{1}$

“…(2) Let V be a Banach space and A a weak * compact absolutely convex subset of V * . Then by [37,Theorem 9], A has the scalar point of continuity property if and only if A is a weak Radon-Nikodým subset (WRN for short). We refer to [3,41,37] for exact definitions and additional information about WRN subsets.…”

A dual geometric characterization of Banach spaces not containingl1

“…If the unit ball of E has the RNP (resp., the WRNP) we say that E has the RNP (resp., the WRNP). For more about RNP and WRNP we refer the reader to [4], [8], [5] and [12].…”