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

Abstract:

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