The representation of σ-algebras

Abstract

We know that every Boolean algebra is isomorphic to a field, whereas a complete Boolean algebra need not be isomorphic to a complete field (since, for instance, it need not be atomic). It is natural to ask the intermediate question: is every σ-algebra isomorphic to a σ-field? The answer is no. We shall see, in fact, that if A is a non-atomic σ-algebra satisfying the countable chain condition, then A cannot be isomorphic to a σ-field. For an example of such an algebra consider the regular open algebra of a Hausdorff space with no isolated points and with a countable base. Alternatively, consider either the reduced Borel algebra or the reduced measure algebra of the unit interval.