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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 acomplete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, weobtain a noncommutative version of this result.

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