The representation of abstract measure functions

Abstract

Marchthat X2(0(xi)) =X2(<£(yi)) if and only if Xi(xi) =Xi(yi).If we further have X2((/>(xi)) =Xi(xi) for a suitable isomorphism 4>, then (£1, Xi) and (E2, X2) are said to be isometric.We shall here be interested in abstract measure algebras only to within isomorphism;and this permits a further reduction of the problem.In (£, X) we define the "induced" equivalence relation, ~, by writing x~y to mean X(x) =X(y).The properties of (E, X) to within isomorphism are evidently determined completely by this equivalence relation.Accordingly we define an abstract measure algebra to be either an ordered pair (E, X) as before, or an ordered pair (E, ~), where E is a Boolean cr-algebra and ~ is an equivalence relation on £ satisfying the postulates to be given below.If ~ is the equivalence relation induced by X, we say that (E, X) and (E, ~) are naturally isomorphic.The définition of isomorphism between two abstract measure algebras is now clear; for example, (£1, ~) and (£2, ~) will be isomorphic if and only if there exists a mapping ip oí Ei on E2 such that (i) tp is an algebraic isomorphism and (ii) 4>ixi) ~4>iyi) if and only if Xi~yi.