On the isomorphism problem for measures on Boolean algebras

The paper investigates possible generalizations of Maharam’s theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure. We give some partial analogues to be used for a classification of algebras that support a finitely additive non-atomic measure with a higher uniform regularity number. We show that some naturally induced equivalence relations connected to metric isomorphism are quite complex even on the Cantor algebra and therefore probably we cannot hope for a nice general classification theorem for finitely-additive measures. sent an example of a Boolean algebra which supports only separable measures but no uniformly regular one.