Geodesic
Ergodic measures and the definability of subgroups via normal extensions of such measures
Teoriâ slučajnyh processov, Tome 18 (2012) no. 1, pp. 58-64

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It is shown that any subgroup $H$ of an uncountable $\sigma$-compact locally compact topological group $\Gamma$ is completely determined by a certain family of left $H$-invariant extensions of the left Haar measure $\mu$ on $\Gamma$. An abstract analogue of this fact is also established for a nonzero $\sigma$-finite ergodic measure given on an uncountable commutative group.
Keywords: Locally compact topological group, Haar measure, invariant extension of measure, ergodicity, commutative group. 
A. B. Kharazishvili. Ergodic measures and the definability of subgroups via normal extensions of such measures. Teoriâ slučajnyh processov, Tome 18 (2012) no. 1, pp. 58-64. http://geodesic.mathdoc.fr/item/THSP_2012_18_1_a2/
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[1] V. Bogachev, Measure Theory, Springer-Verlag, Berlin-Heidelberg, 2007 | MR | Zbl

[2] V. V. Buldygin, A. B. Kharazishvili, Geometric Aspects of Probability Theory and Mathematical Statistics, Kluwer Academic Publishers, Dordrecht, 2000 | MR | Zbl

[3] W. W. Comfort, “Topological groups”, Handbook of Set-Theoretic Topology, eds. K. Kunen, J. E. Vaughan, North-Holland Publ. Co., Amsterdam, 1984 | MR

[4] P. R. Halmos, Measure Theory, D. Van Nostrand, New York, 1950 | MR | Zbl

[5] E. Hewitt, K. A. Ross, Abstract Harmonic Analysis, v. I, Springer-Verlag, Berlin, 1963 | Zbl

[6] T. Jech, Set Theory, Academic Press, New York-London, 1978 | MR | Zbl

[7] S. Kakutani, J. C. Oxtoby, “Construction of a nonseparable invariant extension of the Lebesgue measure space”, Ann. Math., 52 (1950), 580–590. | DOI | MR | Zbl

[8] A. Kanamori, The Higher Infinite, Springer-Verlag, Heifdelberg, 2003 | MR | Zbl

[9] A. B. Kharazishvili, Invariant Extensions of the Lebesgue Measure, Tbilisi State University Press, Tbilisi, 1983 (in Russian) | MR

[10] A. B. Kharazishvili, “To the uniqueness property of the Haar measure”, Reports of Seminar of I. Vekua, 18, Institute of Applied Mathematics TSU, 1984 (in Russian) | MR

[11] A.Ḃ. Kharazishvili, Nonmeasurable Sets and Functions, Elsevier, Amsterdam, 2004 | MR | Zbl

[12] A. B. Kharazishvili, Topics in Measure Theory and Real Analysis, Atlantis Press and World Scientific, Amsterdam-Paris, 2009 | MR | Zbl

[13] K. Kodaira, S. Kakutani, “A nonseparable translation invariant extension of the Lebesgue measure space”, Ann. Math., 52 (1950), 574–579 | DOI | MR | Zbl

[14] K. Kunen, Set Theory, North-Holland Publ. Co., Amsterdam, 1980 | MR | Zbl

[15] K. Kuratowski, A. Mostowski, Set Theory, North-Holland Publ. Co., Amsterdam, 1967 | MR

[16] J. C. Oxtoby, Measure and Category, Springer-Verlag, New York, 1971 | MR | Zbl

[17] J. C. Oxtoby, S. Ulam, “Measure-preserving homeomorphisms and metrical transitivity”, Annals of Mathematics, 42:2 (1941), 874–920 | DOI | MR | Zbl

[18] Sh. Pkhakadze, “The theory of Lebesgue measure”, Trudy Tbilis. Mat. Inst. im. A. Razmadze, 25, Akad. Nauk Gruz. SSR, 1958, 3–272 (in Russian) | MR

[19] A. V. Skorokhod, Integration in Hilbert space, Springer-Verlag, Berlin-Heidelberg, 1974 | MR | Zbl