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

A theorem of Lyapunov states that the range R(μ) of a nonatomic vector measure μ is compact and convex. In this paper we give a condition to detect the dimension of the extremal faces of R(μ) in terms of the Radon-Nikodym derivative of μ with respect to its total variation μ: namely, R(μ) has an extremal face of dimension less than or equal tokif and only if the set (x1,...,xk+1) such thatf(x1),...,f(xk+1) are linear dependent has positive μ⊗(k+1)-measure. Decomposing the setXin a suitable way, we obtain R(μ) as a vector sum of sets which are strictly convex. This result allows us to study the problem of the description of the range of μ if μ has atoms, achieving an extension of Lyapunov's theorem.