Ergodic measures and the definability of subgroups via normal extensions of such measures

A. B. Kharazishvili

A. Razmadze Mathematical Institute, University Street, 2, Tbilisi 0186, Georgia

Abstract: 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.

MSC: 28A20, 28D05, 22B99

Language: English

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