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. 
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     author = {A. B. Kharazishvili},
     title = {Ergodic measures and the definability of subgroups via normal extensions of such measures},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {58--64},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2012_18_1_a2/}
}
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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/