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Strictly positive measures on Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Mirna Džamonja  and
Grzegorz Plebanek
Mirna Džamonja
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR47TJ, UK, E-mail: [email protected]
Grzegorz Plebanek
Affiliation:
Institute of Mathematics, Wrocław University, 2/4 PL. Grunwaldzki, 50-384 Wrocław, Poland, E-mail: [email protected]
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Abstract

We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA + ¬ CH every atomless ccc Boolean algebra satisfies this stronger chain condition.

Keywords

strictly positive measurechain conditions
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1416 - 1432
Copyright
Copyright © Association for Symbolic Logic 2008

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