Geodesic
Classification of measurable functions of several variables and matrix distributions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 4, pp. 46-59

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We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, this is an invariant of this function with respect to a certain group of transformations of variables; on the other hand, this is a special probability measure in the space of matrices (tensors) that is invariant under actions of natural infinite permutation groups. The intricate interplay of both interpretations of matrix (tensor) distributions makes them an important subject of modern functional analysis. We formulate and prove a theorem that, under certain conditions on a measurable function of two variables, its matrix distribution is a complete invariant.
Keywords: classification of functions, metric triples, pointwise ergodic theorem.
Mots-clés : matrix distribution 
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     author = {A. M. Vershik},
     title = {Classification of measurable functions of several variables and matrix distributions},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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     year = {2023},
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A. M. Vershik. Classification of measurable functions of several variables and matrix distributions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 4, pp. 46-59. http://geodesic.mathdoc.fr/item/FAA_2023_57_4_a3/