The Journal of Symbolic Logic

Research Article

Strictly positive measures on Boolean algebras

Mirna Džamonjaa1 and Grzegorz Plebaneka2

a1 School of Mathematics, University of East Anglia, Norwich NR47TJ, UK, E-mail: h020@uea.ac.uk

a2 Institute of Mathematics, Wrocław University, 2/4 PL. Grunwaldzki, 50-384 Wrocław, Poland, E-mail: grzes@math.uni.wroc.pl

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.

(Received February 15 2008)

Key words and phrases

  • strictly positive measure;
  • chain conditions