Symmetric Measures on Cartesian Products

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