Let $G$ be a group acting on a probability measure space $(X, \mu)$ by measure-preserving transformations. I have read the two following definitions of ergodicity of such an action:
- For every measurable set $A$ such that $\mu(gA \Delta A) = 0$ for all $g \in G$, we have $\mu(A) = 0$ or $1$.
- For every measurable set $A$ such that $gA = A$ for all $g \in G$, we have $\mu(A) = 0$ or $1$.
It is clear that 1 implies 2. It is not hard to show that 2 implies 1 if the group is countable. But what about the more general case? What if, for instance, $G$ is a separable group acting continuously on a topological, or even metric space $X$? Would that be enough?
Also, it is clear that a transitive action satisfies 2. But when does it satisfy 1 in the general case? Would the hypotheses above help?