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Classification of measurable functions of several variables and invariantly distributed random matrices. (English. Russian original) Zbl 1025.28010

Funct. Anal. Appl. 36, No. 2, 93-105 (2002); translation from Funkts. Anal. Prilozh. 36, No. 2, 12-27 (2002).
Summary: 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.

MSC:

28D05 Measure-preserving transformations
60E05 Probability distributions: general theory
28A35 Measures and integrals in product spaces
15B52 Random matrices (algebraic aspects)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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