Compactness of the congruence group of measurable functions in several variables

Abstract

We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let \(f:\prod\limits_{i = 1}^n {X_i \to Z} \) be a measurable function defined on the product of finitely many standard probability spaces (Xi, \(\mathfrak{B}_i \), μi), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g1, ..., gn) of measure preserving automorphisms of the respective spaces (Xi, \(\mathfrak{B}_i \), μi) such that f(g1 x 1, ..., gnxn) = f(x1, ..., xn) almost everywhere and prove that this group is compact, provided that its “trivial” symmetries are factored out. As a consequence, we are able to characterize all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which was solved by the first author but is interesting in itself. Bibliography: 5 titles.