Essentially free action

A measurable action of a countable discrete group $G$ on a standard probability space is essentially free if

\mu\{x:T_gx=x\}=0\qquad(g\ne e).

Since $G$ is countable, this is equivalent to almost every point having trivial stabilizer.

Examples

An irrational circle rotation gives an essentially free $\mathbb Z$ -action: no nonzero power fixes a point. A cycle of length $n$ is ergodic as a $\mathbb Z$ -system but not essentially free, since $T^n$ is the identity. The action of $\mathbb Z/n\mathbb Z$ by translation on itself is free; the acting group matters.