Geodesic
On the classification problem of measurable functions in several variables and on matrix distributions
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 119-143 
A. M. Vershik; U. Haböck. On the classification problem of measurable functions in several variables and on matrix distributions. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 119-143. http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a7/
@article{ZNSL_2015_441_a7,
     author = {A. M. Vershik and U. Hab\"ock},
     title = {On the classification problem of measurable functions in several variables and on matrix distributions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {119--143},
     year = {2015},
     volume = {441},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a7/}
}
TY  - JOUR
AU  - A. M. Vershik
AU  - U. Haböck
TI  - On the classification problem of measurable functions in several variables and on matrix distributions
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 119
EP  - 143
VL  - 441
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a7/
LA  - en
ID  - ZNSL_2015_441_a7
ER  - 
%0 Journal Article
%A A. M. Vershik
%A U. Haböck
%T On the classification problem of measurable functions in several variables and on matrix distributions
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 119-143
%V 441
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a7/
%G en
%F ZNSL_2015_441_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We resume the results from [12] on the classification of measurable functions in several variables, with some minor corrections of purely technical nature. We give a partial solution of he characterization problem of so-called matrix distributions, which are the metric invariants of measurable functions introduced in [12]. Matrix distibutions considered as $\S_\mathbb N\times\S_\mathbb N$-invariant, ergodic measures on the space of matrices – this fact connects our problem with Aldous' and Hoover's theorem [2,6].

[1] D. J. Aldous, “Exchangability and related topics”, École d'Été de Probabilités de Saint-Flour XIII – 1983, Lecture Notes Math., 1117, Springer, 2006, 1–198 | DOI | MR

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

[3] V. I. Bogachev, Measure Theory, v. I, Springer, Berlin–Heidelberg, 2007 | MR | Zbl

[4] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, 152, Birkhäuser Boston, Boston, MA, 1999 | MR | Zbl

[5] U. Haböck, Cohomology and Classifications Problems in Dynamics, PhD Thesis, Faculty of Mathematics, University of Vienna, 2006

[6] D. N. Hoover, “Row-column exchangeability and a generalized model for exchangeability”, Exchangeability in Probability and Statistics, eds. G. Koch, F. Spizzichino, North-Holland, Amsterdam, 1982, 281–291 | MR

[7] V. A. Rokhlin, “Metric classification of measurable functions”, Uspekhi Mat. Nauk, 12:2(74) (1957), 169–174 | MR | Zbl

[8] A. M. Vershik, Invariant measures – new aspects of dynamics, combinatorics and representation theory, Takagi Lectures, Math. Inst. of Tōhoku University, 2015

[9] A. M. Vershik, “On classification of measurable functions of several variables”, Zapiski Nachn. Semin. POMI, 403, 2012, 35–57 | MR

[10] A. M. Vershik, U. Haböck, “Compactness of the congruence group of measurable functions in several variables”, J. Math. Sciences, 141:6 (2007), 1601–1607 | DOI | MR

[11] A. M. Vershik, “Random metric spaces and universality”, Russian Math. Surveys, 59:2 (2004), 259–295 | DOI | MR | Zbl

[12] A. M. Vershik, “Classification of measurable functions of several arguments, and invariantly distributed random matrices”, Funkt. Anal. Prilozhen., 36:2 (2002), 12–27 | DOI | MR | Zbl

[13] A. M. Vershik, “A random metric space is a Urysohn space”, Dokl. Akad. Nauk, 387:6 (2002), 733–736 | MR | Zbl

[14] A. M. Vershik, “Description of invariant measures for the actions of some infinite-dimensional groups”, Dokl. Akad. Nauk SSSR, 218:4 (1974), 749–752 | MR | Zbl