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Algebraic characterizations of measure algebras


Author: Thomas Jech
Journal: Proc. Amer. Math. Soc. 136 (2008), 1285-1294
MSC (2000): Primary 28A60, 06E10
DOI: https://doi.org/10.1090/S0002-9939-07-09184-8
Published electronically: December 28, 2007
MathSciNet review: 2367102
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Abstract: We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean $ \sigma$-algebra. For instance, a Boolean $ \sigma$-algebra $ B$ is a measure algebra if and only if $ B-\{\boldsymbol{0}\}$ is the union of a chain of sets $ C_1\subset C_2\subset ...$ such that for every $ n$,

(i)
every antichain in $ C_n$ has at most $ K(n)$ elements (for some integer $ K(n)$),
(ii)
if $ \{a_n\}_n$ is a sequence with $ a_n \notin C_n$ for each $ n$, then $ \lim_n a_n =\boldsymbol{0}$, and
(iii)
for every $ k$, if $ \{a_n\}_n$ is a sequence with $ \lim_n a_n =\myzero$, then for eventually all $ n$, $ a_n \notin C_k$.
The chain $ \{C_n\}$ is essentially unique.


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Additional Information

Thomas Jech
Affiliation: Mathematical Institute, AS CR, Zitna 25, CZ - 115 67 Praha 1, Czech Republic
Email: jech@math.cas.cz

DOI: https://doi.org/10.1090/S0002-9939-07-09184-8
Received by editor(s): December 11, 2006
Published electronically: December 28, 2007
Additional Notes: This work was supported in part by GAAV Grant IAA100190509
Communicated by: David Preiss
Article copyright: © Copyright 2007 American Mathematical Society

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