Symmetric measures on Cartesian products

Abstract

This paper has its origin in the theory of probability, but we think it may be of interest to some who are not familiar with probabilistic technique and jargon.Accordingly, we use for the most part the language of measure theory instead of the language of probability.However, an informal probabilitistic statement of the problem will, we hope, pave the way for all readers.Suppose that, for each value ir of a parameter, {en}^°_i is a sequence of random variables that are statistically independent and subject to a common distribution depending on ir.If now ir itself is a random variable, consider the over-all distribution of the sequence {e"j"_i, i.e., the average with respect to ir of the conditional distribution of the sequence given ir.This overall distribution will not in general render the en's independent, as the conditional distributions given w are assumed to do.Nonetheless, it will obviously, like a distribution with independent en's, be invariant under finite permutations of the variables en among themselves, or symmetric, as we shall say.Conversely, it is true under very general circumstances that any symmetric distribution on the e"'s can be constructed from a suitable family of independent distributions, parametrized say by ir, and a suitable distribution of ir.Jules Haag seems to have been the first author to discuss symmetric sequences of random variables (see [13]).This paper deals only with 2-valued random variables.It hints at, but does not rigorously state or prove, the representation theorem for this case.Somewhat later, symmetric distributions were independently introduced by de Finetti: and the representation theorem for symmetric distributions was proved and exploited by him, especially in connection with the foundations of probability, first in case the en's are 2-valued random variables [10; ll].This case has also been treated by Hincin, in much the same manner as by de Finetti [18;19].De Finetti also proved the theorem for the case in which the en's are real random variables [ll].This case has also been treated by Dynkin [9], apparently without knowledge of de Finetti's work in [ll].Dynkin's technique also applies to random variables on all spaces that are, in a certain sense, separable.A few