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 non–atomic 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 or equal to k if and only if the set (x1, . . . , xk+1) such that f(x1), . . . , f(xk+1) are linear dependent has positive |μ|⊗(k+1)–measure. Decomposing the set X in a suitable way, we obtain R(μ) as 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. 1991 Mathematics Subject Classification. Primary: 46G10, 28B05. Secondary: 52A20.