December 2008 Strictly positive measures on Boolean algebras
Mirna Džamonja, Grzegorz Plebanek
J. Symbolic Logic 73(4): 1416-1432 (December 2008). DOI: 10.2178/jsl/1230396929


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+\neg 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+\neg CH every atomless ccc Boolean algebra satisfies this stronger chain condition.


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Mirna Džamonja. Grzegorz Plebanek. "Strictly positive measures on Boolean algebras." J. Symbolic Logic 73 (4) 1416 - 1432, December 2008.


Published: December 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1158.03036
MathSciNet: MR2467227
Digital Object Identifier: 10.2178/jsl/1230396929

Primary: 28E15 , 54G20 , Primary 03E75

Keywords: chain conditions , strictly positive measure

Rights: Copyright © 2008 Association for Symbolic Logic


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Vol.73 • No. 4 • December 2008
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