If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, then $(2^G, \mu)$ is a Borel, essentially free, probability measure preserving action of $G$ on a standard Lebesgue space.


My question is whether there is any analogue of this for locally compact Polish groups. More precisely, if $G$ is a locally compact Polish group, does $G$ admit a Borel, essentially free, probability measure preserving action on a standard Lebesgue space?



Locally compact Polish groups acting on standard Lebesgue spaces
Yes - this is Proposition 1.2 of [Adams, Elliott and Giordano](https://www.ams.org/journals/tran/1994-344-02/S0002-9947-1994-1250814-5/) ([MathSciNet review](https://mathscinet.ams.org/mathscinet-getitem?mr=1250814)), Amenable actions of groups, Trans. Amer. Math. Soc. 344 (1994), 803-822.


