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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
DOI: https://doi.org/10.1090/S0002-9947-1981-0617547-8
Erratum: Trans. Amer. Math. Soc. 272 (1982), 809.
MathSciNet review: 617547
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0617547-8
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

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