Descriptive properties of elements of biduals of Banach spaces

If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball B-E* that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of B-E*, generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire classes of L-1-preduals which were introduced by S. A. Argyros, G. Godefroy and H. P. Rosenthal (2003). Also, several examples witnessing natural limits of our positive results are presented.