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.[ Full text: (.ps.gz)]