# Extreme points of the set of elements majorised by an integrable function: Resolution of a problem by Luxemburg and of its noncommutative counterpart

## TL;DR

This paper characterizes the extreme points of the set of functions majorized by an integrable function, solving a long-standing problem by Luxemburg and extending the result to a noncommutative setting.

## Contribution

It provides a complete characterization of extreme points in the majorization set and extends the classical result to noncommutative spaces.

## Key findings

- Complete description of extreme points of majorized functions
- Resolution of Luxemburg's 1967 problem
- Extension to noncommutative integration theory

## Abstract

Let $f$ be an arbitrary integrable function on a finite measure space $(X,\Sigma, \nu)$. We characterise the extreme points of the set $\Omega (f)$ of all measurable functions on $(X,\Sigma, \nu)$ majorised by $f$, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.