Abstract
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.
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References
D. J. Aldous, “Representations for partially exchangeable arrays of random variables,” J. Multivariate Anal., 11, No.4, 581–598 (1981).
D. Aldous, “Exchangeability and related topics,” In: École d'ètè de probabilitès de Saint-Flour, XIII–1983, Lect. Notes in Math., 1117, Springer-Verlag, Berlin, 1985, pp. 1–198.
J. D. Clemens, S. Gao, and A. S. Kechris, “Polish metric spaces: their classification and isometry groups,” Bull. Symbolic Logic, 7, No.3, 361–375 (2001).
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhauser, 1999.
O. Kallenberg, “Symmetries on random arrays and set-indexed processes,” J. Theoret. Probab., 5, No.4, 727–765 (1992).
O. Kallenberg, “Random arrays and functionals with multivariate rotational symmetries,” Probab. Theory Related Fields, 103, No.1, 91–141 (1995).
G. Olshanski and A. Vershik, “Ergodic unitarily invariant measures on the space of infinite Hermitian matrices,” Amer. Math. Soc. Transl. Ser. 2, Vol. 175, 137–175 (1996).
D. Pickrell, “Separable representations for automorphism groups of infinite symmetric spaces,” J. Funct. Anal., 90, 1–26 (1990).
V. A. Rokhlin, “Metric classification of measurable functions,” Usp. Mat. Nauk, 12, no.2 (74), 169–174 (1957).
A. M. Vershik, “The universal Uryson space, Gromov's metric triples, and random metrics on the series on natural numbers,” Usp. Mat. Nauk, 53, No.5, 57–64 (1998).
A. M. Vershik, “A description of invariant measures for actions of certain infinite-dimensional groups,” Dokl. Akad. Nauk SSSR, 218, No.4, 749–752 (1974).
A. Vershik, “Classification of the Measurable Functions of Several Variables and Invariant Measures on the Space of Matrices and Tensors,” Preprint ESI (The Erwin Schrödinger International Institute for Mathematical Physics), No. 1107 (2001), pp. 1–23.
A. Vershik, “Random metric spaces and the universal Urysohn space,” MSRI Preprint No. 2002-018; In: Fundamental Mathematics Today: 10 anniversary of Independent Moscow University, MCCME Publ., M., 2002.
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Vershik, A.M. Classification of Measurable Functions of Several Variables and Invariantly Distributed Random Matrices. Functional Analysis and Its Applications 36, 93–105 (2002). https://doi.org/10.1023/A:1015662321953
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DOI: https://doi.org/10.1023/A:1015662321953