On Boolean algebras with strictly positive measures

We investigate reflection-type problems on the class SPM, of Boolean algebras carrying strictly positive finitely additive measures. We show, in particular, that in the constructible universe there is a Boolean algebra $\mathfrak{A}$ which is not in SPM but every subalgebra of $\mathfrak{A}$ of cardinality $\mathfrak{c}$ admits a strictly positive measure. This result is essentially due to Farah and Velickovic.