## Journal of Symbolic Logic

### Strictly positive measures on Boolean algebras

#### 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+\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.

#### Article information

Source
J. Symbolic Logic Volume 73, Issue 4 (2008), 1416-1432.

Dates
First available in Project Euclid: 27 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1230396929

Digital Object Identifier
doi:10.2178/jsl/1230396929

Mathematical Reviews number (MathSciNet)
MR2467227

Zentralblatt MATH identifier
1158.03036

#### Citation

Džamonja, Mirna; Plebanek, Grzegorz. Strictly positive measures on Boolean algebras. J. Symbolic Logic 73 (2008), no. 4, 1416--1432. doi:10.2178/jsl/1230396929. http://projecteuclid.org/euclid.jsl/1230396929.