A group or group action is amenable if it has an invariant mean. Apparently, the term "amenable" is a play of words on "mean" (in British English *amenable* is pronounced "a-mean-able", hinting at "able to support a mean"), see this <A HREF="https://math.stackexchange.com/a/1307539/87355">MSE posting</A>. Measurable (from the original name in German) would have been a more appropriate name, to help intuition.