The extreme points of some convex sets in the theory of majorization

Abstract

Let (A, %plane1D;49C;, μ) be a finite measure space, and let Ω_{µ, w}⁺f denote the set of all nonnegative real-valued %plane1D;49C;-measurable functions on A weaklymajorized by a nonnegative function f, in the sense of Hardly, Littlewood and Pólya. For a nonatomic µ, the extreme points ofΩ_{µ, w} ⁺f are shown to be the nonnegativefunctions obtained by taking a fraction (1−θ) of the largest values of and arranging them in any way on any subset of A of measure(1−θ), with values elsewhere set equal to zero. Topological properties of these extreme points are given.