Geodesic
Transformations which leave a measure quasi-invariant
Matematičeskie zametki, Tome 7 (1970) no. 2, pp. 223-227

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It is shown that every countable group $G$ has a faithful representation as an ergodic freely-acting group of transformations of a commutative Neumann algebra $M$ with measure $\mu$, leaving the measure $\mu$ quasi-invariant, while there does not exist a measure $\mu'$ which is equivalent to $\mu$ and invariant with respect to $G$. 
@article{MZM_1970_7_2_a10,
     author = {V. Ya. Golodets},
     title = {Transformations which leave a measure quasi-invariant},
     journal = {Matemati\v{c}eskie zametki},
     pages = {223--227},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_7_2_a10/}
}
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V. Ya. Golodets. Transformations which leave a measure quasi-invariant. Matematičeskie zametki, Tome 7 (1970) no. 2, pp. 223-227. http://geodesic.mathdoc.fr/item/MZM_1970_7_2_a10/