Geodesic
Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 2, pp. 12-27.

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The classification of measurable functions of several variables is reduced to the problem of describing some special measures on the matrix (tensor) space, namely, the so-called matrix (tensor) distributions, that are invariant with respect to the permutations of indices. In the case of functions with additional symmetries (symmetric, unitarily or orthogonally invariant, etc.), these measures also have additional symmetries. This relationship between measurable functions and measures on the tensor space as well as our method in itself are used in both directions, namely, on one hand, to investigate invariance properties of functions and characterizations of matrix distributions, and, on the other hand, to classify the set of all invariant measures. We also give a canonical model of a measurable function with a given matrix distribution.
Keywords: classification of functions, infinite symmetric group.
Mots-clés : matrix distributions 
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A. M. Vershik. Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 2, pp. 12-27. http://geodesic.mathdoc.fr/item/FAA_2002_36_2_a1/

[1] Aldous D., “Representations for partially exchangeable arrays of random variables”, J. Multivariate Anal., 11:4 (1981), 581–598 | DOI | MR | Zbl

[2] Aldous D., “Exchangeability and related topics”, Ecole d'ete de probabilites de Saint Flour, XIII – 1983, Lect. Notes Math., 1117, Springer-Verlag, 1–198 | MR

[3] Clemens J., Gao S., Kechris A., “Polish metric spaces: their classification and isometry groups”, Bull. Symbolic Logic, 7:3 (2001), 361–375 | DOI | MR | Zbl

[4] Gromov M., Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhauser, 1998 | MR | Zbl

[5] Kallenberg O., “Symmetries on random arrays and set-indexed processes”, J. Theoret. Probab., 5:4 (1992), 727–765 | DOI | MR | Zbl

[6] Kallenberg O., “Random arrays and functionals with multivariate rotational symmetries”, Probab. Theory Related Fields, 103:1 (1995), 91–141 | DOI | MR | Zbl

[7] Olshanski G., Vershik A., “Ergodic unitary invariant measures on the space of infinite Hermitian matrices”, Amer. Math. Soc. Transl. Ser. 2, 175, 1996, 137–175 | MR | Zbl

[8] Pickrell D., “Separable representations for automorphism groups of infinite symmetric spaces”, J. Funct. Anal., 90 (1990), 1–26 | DOI | MR | Zbl

[9] Rokhlin V. A., “Metricheskaya klassifikatsiya izmerimykh funktsii”, UMN, 12:2 (1957), 169–174 | MR | Zbl

[10] Vershik A. M., “Universalnye prostranstva Urysona, troiki Gromova i sluchainye metriki na ryadakh naturalnykh chisel”, UMN, 53:5 (1998), 57–64 | DOI | MR | Zbl

[11] Vershik A. M., “Opisanie invariantnykh mer dlya deistvii nekotorykh beskonechnomernykh grupp”, DAN SSSR, 218:4 (1974), 749–752 | Zbl

[12] Vershik A., Classification of the Measurable Functions of Several Arguments and Invariant Measures on the Space of Matrices and Tensors, Preprint ESI (The Erwin Schroedinger International Institute fur Mathematical Physics), No. 1107, 2001, 1–23 | MR

[13] Vershik A., The universe of the metric spaces and the universal Urysohn space, Preprint MPIM #8, 2002, math.GT/0203008. Fundamental Mathematics Today: 10 anniversary of Independent Moscow University, MCCME Publ., M., 2002 | MR