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

Liapounoffs theorem states that if (X, 2) is a measurable space and µ: Ʃ→Rd is nonatomic, bounded, and countably additive, thenR(µ)= {µ(A): A ƐƩ) is compact and convex. When Ʃ is replaced by a a-complete Boolean algebra or an F-algebra (to be defined) and n is allowed to be only finitely additive, is still convex. If Ʃ is any Boolean algebra supporting nontrivial, nonatomic, finitely-additive measures and Z is a zonoid, there exists a nonatomic measure onƩ with range dense in Z. A wide variety of pathology is examined which indicatesthat ranges of finitely-additive, nonatomic, finite-dimensional, vector-valued measures are fairly arbitrary.