Extremal faces of the range of a vector measure and a theorem of Lyapunov

A theorem of Lyapunov states that the range R(mu) of a nonatomic vector mea sure mu is compact and convex. In this paper we give a condition to detect the dimension of the extremal faces of R(mu) in terms of the Radon-Nikodym derivative of mu with respect to its total variation \mu\: namely, R(mu) ha s an extremal face of dimension less than or equal to k if and only if the set (x(1),...,x(k+1)) such that f(x(1)),...,f(x(k+1)) are linear dependent has positive \mu\(X(k+1))-measure. Decomposing the set X in a suitable way, we obtain R(mu) as a Vector sum of sets which are strictly convex. This re sult allows us to study the problem of the description of the range of FL i f FL has atoms, achieving an extension of Lyapunov's theorem. (C) 1999 Acad emic Press.