Source: J. Symbolic Logic
Volume 73, Issue 4
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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