## 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