Compactness of the congruence group of measurable functions in several variables

We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let be a measurable function defined on the product of finitely many standard probability spaces (X_i, , μ_i), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g₁, ..., g_n) of measure preserving automorphisms of the respective spaces (X_i, , μ_i) such that f(g₁ x ₁, ..., g_nx_n) = f(x₁, ..., x_n) 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. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 57–67.