Geodesic
Compactness of the congruence group of measurable functions in several variables
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIX, Tome 334 (2006), pp. 57-67 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. M. Vershik and U. Hab\"ock},
     title = {Compactness of the congruence group of measurable functions in several variables},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_334_a3/}
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A. M. Vershik; U. Haböck. Compactness of the congruence group of measurable functions in several variables. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIX, Tome 334 (2006), pp. 57-67. http://geodesic.mathdoc.fr/item/ZNSL_2006_334_a3/

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