In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows:
[amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined on all subsets.
He defines $SO_2$ to be the group of rotations of the unit circle, which he has used to show that $S_1$, the unit circle, is $SO_2$-paradoxical (as an analogue to the usual non-measurable set defined in the interval $[0,1)$ ). I am taking measure theory this term, but am not sure how to assign a measure to subsets of $SO_2$. Thus, I am not really sure where to start in showing whether or not $SO_2$ is an amenable group.
When I look at the Wikipedia entry about amenable groups, I'm unable to make much more sense of the definition in the context of the material.
Is $SO_2$, the group of rotations of the unit circle, an amenable group? If not, why (so that I may build an intuition for these objects)?