Embeddings of metric Boolean algebras in RN

A Boolean algebra A equipped with a (finitely-additive) positive probability measure m can be turned into a metric space (A,dm), where dm(a,b)=m((a∧¬b)∨(¬a∧b)), for any a,b∈A, sometimes referred to as metric Boolean algebra. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in RN (for a certain N) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra A such that the metric space (At(A),dm) embeds isometrically in RN (with the Euclidean metric).