The structure of separable Dynkin algebras

Abstract

We continue to study abstract Dynkin algebras introduced in [2]. Such algebras form a useful instrument for discussing probabilities in a rather natural context. Abstractness means the absence of a set-theoretic structure of elements in such algebras. We introduce a large useful class of abstract algebras — separable Dynkin algebras and indicate also the simplest example of a non-separable algebra. The property of separability allows us to define appropriate variants of Boolean versions to operations of intersection and union of elements. Such operations are in general only partially defined. Some properties of separable algebras are proved. We use them to receive the standard properties of intersection and union including associativity and distributivity in case the corresponding operations are applicable. The established facts allow us to define Boolean subalgebras in a separable Dynkin algebra and to check the coincidence of our version of definition with a usual one. Finally we formulate and prove the main result about structure of separable Dynkin algebras — we represent them as a set-theoretic union of maximal Boolean subalgebras. The proof uses earlier done preparation and standard scheme of Zorn lemma application. Refs 6.