Transformations which leave a measure quasi-invariant

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 $G$.