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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Liapounoff’s theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures
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by Thomas E. Armstrong and Karel Prikry PDF
Trans. Amer. Math. Soc. 266 (1981), 499-514 Request permission

Erratum: Trans. Amer. Math. Soc. 272 (1982), 809.

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
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • 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
  • MathSciNet review: 617547