Download PDF - Ergodic decomposition of quasi-invariant probability measuresAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Ergodic decomposition of quasi-invariant probability measures

Authors 1, 1

Affiliations

  1. Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Wien, Austria

Abstract

The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasi-invariant under a Borel action of a locally compact second countable group or a discrete nonsingular equivalence relation. In the process we obtain a simultaneous ergodic decomposition of all quasi-invariant probability measures with a prescribed Radon-Nikodym derivative, analogous to classical results about decomposition of invariant probability measures.

Keywords

ENG
ergodic decomposition, nonsingular group actions, nonsingular equivalence relations, quasi-invariant measures

Bibliography

  1. P. Billingsley, Probability and Measure, Wiley, New York, 1979.
  2. R. V. Chacon and D. S. Ornstein, A general ergodic theorem, Illinois J. Math. 4 (1960), 153-160.
  3. J. Feldman, P. Hahn and C. C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. Math. 28 (1978), 186-230.
  4. J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), 289-324.
  5. S. R. Foguel, Ergodic decomposition of a topological space, Israel J. Math. 7 (1969), 164-167.
  6. E. Hopf, On the ergodic theorem for positive linear operators, J. Reine Angew. Math. 205 (1960), 101-106.
  7. A. S. Kechris, Countable sections for locally compact groups, Ergodic Theory Dynam. Systems 12 (1992), 283-295.
  8. J. Kerstan and A. Wakolbinger, Ergodic decomposition of probability laws, Z. Wahrsch. Verw. Gebiete 56 (1981), 399-414.
  9. Yu. I. Kifer and S. A. Pirogov, On the decomposition of quasi-invariant measures into ergodic components, Uspekhi Mat. Nauk 27 (1972), no. 5, 239-240 (in Russian).
  10. N. Lusin, Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris, 1930.
  11. W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, Cambridge, 1981.
  12. K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.
  13. R. R. Phelps, Lectures on Choquet's Theorem, van Nostrand Reinhold, New York, 1966.
  14. A. Ramsay, Virtual groups and group actions, Adv. Math. 6 (1971), 253-322.
  15. A. Ramsay, Subobjects of virtual groups, Pacific J. Math. 87 (1980), 389-454.
  16. K. Schmidt, Cocycles on Ergodic Transformation Groups, MacMillan (India), Delhi, 1977.
  17. K. Schmidt, A probabilistic proof of ergodic decomposition, Sankhyā Ser. A 40 (1978), 10-18.
  18. K. Schmidt, Unique ergodicity for quasi-invariant measures, Math. Z. 167 (1979), 168-172.
  19. H. Shimomura, Ergodic decomposition of quasi-invariant measures, Publ. RIMS Kyoto Univ. 14 (1978), 359-381.
  20. H. Shimomura, Remark to the paper 'Ergodic decomposition of quasi-invariant measures', ibid. 19 (1983), 203-205.
  21. H. Shimomura, Remark to the ergodic decomposition, ibid. 26 (1990), 861-865.
  22. M. L. Sturgeon, The ergodic decomposition of conservative Baire measures, Proc. Amer. Math. Soc. 44 (1974), 141-146.
  23. V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191-220.
  24. J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949), 401-485.
  25. R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984.
Colloquium Mathematicum
2000/84/85/2
Export to PBN, Opens in new tab
Pages:
495-514
Main language of publication
English
Received
1999-09-22
Accepted
1999-10-08
Published
2000
Exact and natural sciences