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 Mat. Sb., 2014, Volume 205, Number 2, Pages 39–70 (Mi msb8202)

Ergodic decomposition for measures quasi-invariant under a Borel action of an inductively compact group

A. I. Bufetovabcde

a Steklov Mathematical Institute of the Russian Academy of Sciences
b Rice University
c A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow
d National Research University "Higher School of Economics"
e Aix-Marseille Université

Abstract: The aim of this paper is to prove ergodic decomposition theorems for probability measures which are quasi-invariant under Borel actions of inductively compact groups as well as for $\sigma$-finite invariant measures. For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by the initial invariant measure.
Bibliography: 21 titles.

Keywords: ergodic decomposition, infinite-dimensional groups, quasi-invariant measure, infinite-dimensional unitary group, measurable decomposition.

 Funding Agency Grant Number Agence Nationale de la Recherche ANR-11-IDEX-0001-02 Alfred P. Sloan Foundation Dynasty Foundation Independent University of Moscow Ministry of Education and Science of the Russian Federation MK-6734.2012.1ÌÄ-2859.2014.1 Russian Academy of Sciences - Federal Agency for Scientific Organizations Russian Foundation for Basic Research 10-01-93115-NTsNIL11-01-0065412-01-3128412-01-3302013-01-12449

DOI: https://doi.org/10.4213/sm8202

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English version:
Sbornik: Mathematics, 2014, 205:2, 192–219

Bibliographic databases:

UDC: 517.938
MSC: 28D15, 37A15

Citation: A. I. Bufetov, “Ergodic decomposition for measures quasi-invariant under a Borel action of an inductively compact group”, Mat. Sb., 205:2 (2014), 39–70; Sb. Math., 205:2 (2014), 192–219

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8202
• https://doi.org/10.4213/sm8202
• http://mi.mathnet.ru/eng/msb/v205/i2/p39

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. I. Bufetov, “Finiteness of ergodic unitarily invariant measures on spaces of infinite matrices”, Ann. Inst. Fourier, 64:3 (2014), 893–907
2. A. I. Bufetov, “Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures”, Izv. Math., 79:6 (2015), 1111–1156
3. Proc. Steklov Inst. Math., 292 (2016), 94–111
4. A. I. Bufetov, “Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. II. Convergence of infinite determinantal measures”, Izv. Math., 80:2 (2016), 299–315
5. A. I. Bufetov, “Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. III. The infinite Bessel process as the limit of the radial parts of finite-dimensional projections of infinite Pickrell measures”, Izv. Math., 80:6 (2016), 1035–1056
6. Y. Qiu, “Ergodic measures on compact metric spaces for isometric actions by inductively compact groups”, Proc. Amer. Math. Soc., 145:4 (2017), 1593–1598
7. Y. Qiu, “Infinite random matrices & ergodic decomposition of finite and infinite Hua-Pickrell measures”, Adv. Math., 308 (2017), 1209–1268
8. A. I. Bufetov, Y. Qiu, “Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields”, Compos. Math., 153:12 (2017), 2482–2533
9. Ya. Qiu, “Ergodic measures on infinite skew-symmetric matrices over non-archimedean local fields”, Group. Geom. Dyn., 13:4 (2019), 1401–1416
10. T. Assiotis, “Hua-pickrell diffusions and feller processes on the boundary of the graph of spectra”, Ann. Inst. Henri Poincare-Probab. Stat., 56:2 (2020), 1251–1283
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