Cardinal algebras and measures invariant under equivalence relations.

Abstract

Introduction. There have been discussions from time to time of abstract the values of which need not be numerical (e.g. [2], [3], [4], [6], [7]). One of the purposes of this paper is to present arguments in favor of the use of cardinal as values for these measures. Cardinal were introduced and developed by A. Tarski in [8]. They have many of the good properties of real numbers and arise naturally in situations like the following: A (pseudo) group G of one-one functions is given with domain and range in a u-ring of sets X2 An equivalence relation between members of 1 is defined as follows: A B iffthere are Ai, Bi E 1[,fi E G for i<co such that Ai n Aj=O=B=i r Bj for i#Oj, A=JUi < . Ai, B=Ui<oo Bi, Ai(Domfi andfi*(Ai)=Bi for all i<co. This is equivalence by countable decomposition. Equivalence relations like have been considered in [2], [3], [4], [6], [7]. When the aim is to obtain a that is faithful to an equivalence relation of this form, the first natural step is to consider the equivalence classes determined by the relation. It happens that these equivalence classes with suitably defined finite and infinite addition form a generalized cardinal algebra. For instance, the measure algebras considered in [4] and [6] are generalized cardinal algebras. My second purpose is to determine in what conditions we can obtain a numerical faithful to the equivalence relation; that is, a countably additive that satisfies the following: (a) The only sets with zero are those that necessarily have to have it. That is, sets that have infinitely many disjoint equivalent sets contained in a set of one. These sets I call negligible. (b) For sets with positive measure, it should be valid that two sets have the same iff they are equivalent. These characteristics are specially important with respect to probability measures where we want to be as faithful as possible to the equal likelihood relation (cf. [3]). Theorem 2.11, below, gives sufficient (and almost necessary) conditions on the equivalence relation to obtain such a measure.