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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 3/2002

. . ฅเ่จช
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This preprint was accepted January 10, 2002

Contact:
` A.M.Vershik `

ABSTRACT:
Classification of the measurable functions of the several arguments
can be reduced to the description of the special measures
in the space of matrices (tensors) -- so called matrix
distributions --
which are invariant with respect to the group of permutations
of indices. In the case of additional symmetries of the functions
(symmetric functions, unitary or orthoganal symmetry etc.)
these measures aslo have additional symmetries. This link
between measurable functions and measures on the space of tensors
as well as our method is useful in both directions: for
investigations of the invariant properties of the functions
and characterization of the matrix distributions from one side,
and for the classification of all invariant measures. We also
give a canonical model of trhe function with given matrix distribution.
\newline
The proofs based on the ergodic theorems and on the analysis
of the action of the powers of symmetric groups -- this gives
a new proofs of the results of D. Aldous, O. Kallenberg, D. Pickrel,
A. Vershik--G. Olshansky. In the same time given classification theorem
for the functions can be applied to the Gromov's problem about
classification of the metric spaces with measures (mm-spaces).
These applications and also related problems on universal
Urysohn's space will be considered elsewhere.
Key words: จงฌฅเจ๋ฅ ไใญชๆจจ, ฌ โเจ็ญ๋ฅ
เ แฏเฅคฅซฅญจ๏, ฃเใฏฏ๋ ฏฎคแโ ญฎขฎช, จญข เจ ญโญ๋ฅ ฌฅเ๋.

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