  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.
