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 $f\colon\prod_{i=1}^n X_i\longrightarrow Z$ be a measurable function defined on the product of finitely many standard probability spaces $(X_i,\frak B_i,\mu_i)$, $1\le i\le n$, that takes values in any standard Borel space $Z$. We consider the Borel group of all $n$-tuples $(g_1,\dots,g_n)$ of measure preserving automorphisms of the respective spaces $(X_i,\frak B_i,\mu_i)$ such that $f(g_1x_1,\dots,g_nx_n)=f(x_1,\dots,x_n)$ almost everywhere and prove that this group is compact, provided that its ‘trivial’ symmetries are factored out. As a consequence, we are able to characterise all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which has been solved in [2] but is interesting in itself.