Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Thomas E. Armstrong and Karel Prikry
Journal: Trans. Amer. Math. Soc. 266 (1981), 499-514
MSC: Primary 28B05; Secondary 28A12, 28A60
Erratum: Trans. Amer. Math. Soc. 272 (1982), 809.
MathSciNet review: 617547
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28B05, 28A12, 28A60

Retrieve articles in all journals with MSC: 28B05, 28A12, 28A60


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0617547-8
PII: S 0002-9947(1981)0617547-8
Keywords: Liapounoff's theorem, finite additivity, $ F$-space, $ F$-algebra, real-valued measurable cardinal, uniform measure, Hahn decomposition, zonoid
Article copyright: © Copyright 1981 American Mathematical Society