Cardinal algebras and measures invariant under equivalence relations.

Introduction.There have been discussions from time to time of "abstract measures" 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 algebras as values for these measures.Cardinal algebras 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 a-ring of sets Jf.An equivalence relation between members of Jf is defined as follows : A^B iff there are A¡, Bi e Cti,fi e G for /<oo such that At n Aj = 0 = Bi n B¡ for 'V/ ^ = Ui<oo At, 5=Ui<« Bt, A^Domfi and ft*(Ai) = Bi for all i