Extremal faces on the range of a vector measure and a theorem of Lyapunov (Q1283013)

Let \({\mathfrak M}\) be a \(\sigma\)-algebra, \(n\in\mathbb{N}\) and \(\mu:{\mathfrak M}\to \mathbb{R}^n\) a \(\sigma\)-additive measure. A well-known theorem of Lyapunov states that the range of \(\mu\) is convex if \(\mu\) is nonatomic. The main result of the paper is an interesting extension of Lyapunov's theorem; it contains a characterization of the measures \(\mu:{\mathfrak M}\to \mathbb{R}^n\) with convex range. (Reviewer's remark: The main result was already obtained by \textit{R. Herschbach} [Math. Nachr. 181, 215-229 (1996; Zbl 0863.28004)]. For the case \(n= 1\) see \textit{S. Marcus} [Acad. Républ. Popul. Roum., Rev. Math. Pur. Appl. 7, 327-332 (1962; Zbl 0135.26001)]).