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.
@article{bwmeta1.element.bwnjournal-article-cmv84i2p495bwm,
author = {Gernot Greschonig and Klaus Schmidt},
title = {Ergodic decomposition of quasi-invariant probability measures},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {495-514},
zbl = {0972.37003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p495bwm}
}
Greschonig, Gernot; Schmidt, Klaus. Ergodic decomposition of quasi-invariant probability measures. Colloquium Mathematicae, Tome 84/85 (2000) pp. 495-514. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p495bwm/
[000] [1] P. Billingsley, Probability and Measure, Wiley, New York, 1979. | Zbl 0411.60001
[001] [2] R. V. Chacon and D. S. Ornstein, A general ergodic theorem, Illinois J. Math. 4 (1960), 153-160. | Zbl 0134.12102
[002] [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. | Zbl 0392.28023
[003] [4] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), 289-324. | Zbl 0369.22009
[004] [5] S. R. Foguel, Ergodic decomposition of a topological space, Israel J. Math. 7 (1969), 164-167. | Zbl 0179.08302
[005] [6] E. Hopf, On the ergodic theorem for positive linear operators, J. Reine Angew. Math. 205 (1960), 101-106. | Zbl 0103.33801
[006] [7] A. S. Kechris, Countable sections for locally compact groups, Ergodic Theory Dynam. Systems 12 (1992), 283-295.
[007] [8] J. Kerstan and A. Wakolbinger, Ergodic decomposition of probability laws, Z. Wahrsch. Verw. Gebiete 56 (1981), 399-414. | Zbl 0444.60004
[008] [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). | Zbl 0248.28014
[009] [10] N. Lusin, Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris, 1930. | Zbl 56.0085.01
[010] [11] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, Cambridge, 1981. | Zbl 0449.28016
[011] [12] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. | Zbl 0153.19101
[012] [13] R. R. Phelps, Lectures on Choquet's Theorem, van Nostrand Reinhold, New York, 1966. | Zbl 0135.36203
[013] [14] A. Ramsay, Virtual groups and group actions, Adv. Math. 6 (1971), 253-322. | Zbl 0216.14902
[014] [15] A. Ramsay, Subobjects of virtual groups, Pacific J. Math. 87 (1980), 389-454. | Zbl 0453.22005
[015] [16] K. Schmidt, Cocycles on Ergodic Transformation Groups, MacMillan (India), Delhi, 1977.
[016] [17] K. Schmidt, A probabilistic proof of ergodic decomposition, Sankhyā Ser. A 40 (1978), 10-18. | Zbl 0412.60004
[017] [18] K. Schmidt, Unique ergodicity for quasi-invariant measures, Math. Z. 167 (1979), 168-172. | Zbl 0416.28013
[018] [19] H. Shimomura, Ergodic decomposition of quasi-invariant measures, Publ. RIMS Kyoto Univ. 14 (1978), 359-381. | Zbl 0391.60004
[019] [20] H. Shimomura, Remark to the paper 'Ergodic decomposition of quasi-invariant measures', ibid. 19 (1983), 203-205. | Zbl 0509.60008
[020] [21] H. Shimomura, Remark to the ergodic decomposition, ibid. 26 (1990), 861-865. | Zbl 0716.28005
[021] [22] M. L. Sturgeon, The ergodic decomposition of conservative Baire measures, Proc. Amer. Math. Soc. 44 (1974), 141-146. | Zbl 0285.28020
[022] [23] V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191-220. | Zbl 0192.14203
[023] [24] J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949), 401-485. | Zbl 0034.06102
[024] [25] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984. | Zbl 0571.58015