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

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.