Locally compact Polish groups acting on standard Lebesgue spaces

Question

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?

Answer 1

Yes - this is Proposition 1.2 of Adams, Elliott and Giordano (MathSciNet review), Amenable actions of groups, Trans. Amer. Math. Soc. 344 (1994), 803-822.

Answer 2

I think that the analogue of a Bernoulli shift is rather the translation action on a Poisson point process. It appears in the work of Ornstein and Weiss on entropy theory of amenable groups as the analogue of the Bernoulli shift for locally compact groups and a more general form of it appears in the book of Kornfeld Sinai and Fomin under the name of Poisson suspensions.

An informal description of the model is as follows. Let $\Pi_m$ a Poisson point process over $G$ with the Haar measure $m$ as the intensity measure. This can be viewed as a measure on the countable discrete subsets of $G$ and the action of $G$ is by translating the whole configuration.That is for $\nu\subset G$, $g\nu=(g^{-1}h)_{h\in\nu}$. This action is free, ergodic and measure-preserving.