Resolution of a problem by W.A.J. Luxemburg

Abstract: 

In 1967, W.A.J. Luxemburg raised a problem: Determine all the extreme points of the set of elements majorised by an integrable function on an arbitrary finite measure space. The atomless case has been resolved by Ryff. However, the general case is still open. During the past decades, several authors have worked on the non-commutative version of Ryff's result. We provide a complete answer to the problem raised by Luxemburg, which unifies Ryff's result and the classic result for vectors with significant  extension. Moreover, we obtain a noncommutative version of this result.