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

Let $f$ be an arbitrary integrable function on a finite measure space $(X,Σ, ν)$. We characterise the extreme points of the set $Ω(f)$ of all measurable functions on $(X,Σ, ν)$ 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.

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