Liapounoff’s theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures

Abstract

Liapounoff’s theorem states that if ( X , Σ ) (X,\Sigma ) is a measurable space and μ : Σ → R d \mu :\Sigma \to {{\mathbf {R}}^d} is nonatomic, bounded, and countably additive, then R ( μ ) = { μ ( A ) : A ∈ Σ } \mathcal {R}(\mu ) = \{ \mu (A):A \in \Sigma \} is compact and convex. When Σ \Sigma is replaced by a σ \sigma -complete Boolean algebra or an F F -algebra (to be defined) and μ \mu is allowed to be only finitely additive, R ( μ ) \mathcal {R}(\mu ) is still convex. If Σ \Sigma is any Boolean algebra supporting nontrivial, nonatomic, finitely-additive measures and Z Z is a zonoid, there exists a nonatomic measure on Σ \Sigma with range dense in Z Z . A wide variety of pathology is examined which indicates that ranges of finitely-additive, nonatomic, finite-dimensional, vector-valued measures are fairly arbitrary.