On the isomorphism problem for measures on Boolean algebras

TL;DR

This paper explores the classification of Boolean algebras supporting finitely additive measures, extending Maharam's theorem, and reveals the complexity of metric isomorphism relations among such algebras.

Contribution

It proves that Boolean algebras with non-atomic uniformly regular measures are metrically isomorphic to subalgebras of the Jordan algebra with Lebesgue measure, and discusses the complexity of classifying these algebras.

Findings

Boolean algebras with non-atomic uniformly regular measures are isomorphic to subalgebras of the Jordan algebra.

Some equivalence relations related to metric isomorphism are highly complex.

An example of a Boolean algebra supports only separable measures but no uniformly regular measure.

Abstract

The paper investigates possible generalisations 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. We present an example of a Boolean algebra which supports only separable measures but no uniformly regular one.