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 is a measurable space and is nonatomic, bounded, and countably additive, then is compact and convex. When is replaced by a -complete Boolean algebra or an -algebra (to be defined) and is allowed to be only finitely additive, is still convex. If is any Boolean algebra supporting nontrivial, nonatomic, finitely-additive measures and is a zonoid, there exists a nonatomic measure on with range dense in . 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,
-space,
-algebra,
real-valued measurable cardinal,
uniform measure,
Hahn decomposition,
zonoid

Article copyright:
© Copyright 1981
American Mathematical Society