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

Thomas E. Armstrong, Karel Prikry

Research output: Contribution to journalArticle

22 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)499-514
Number of pages16
JournalTransactions of the American Mathematical Society
Volume266
Issue number2
DOIs
StatePublished - Jan 1 1981

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