TY - JOUR

T1 - Liapounoff’s theorem for nonatomic, finitely -additive, bounded, finite-dimensional, vector-valued measures

AU - Armstrong, Thomas E.

AU - Prikry, Karel

PY - 1981/1/1

Y1 - 1981/1/1

N2 - 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.

AB - 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.

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U2 - 10.1090/S0002-9947-1981-0617547-8

DO - 10.1090/S0002-9947-1981-0617547-8

M3 - Article

AN - SCOPUS:0001427017

VL - 266

SP - 499

EP - 514

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -