Abstract
We show that under certain general conditions any finitely additive measure which is defined for all subsets of a set $X$ and is invariant under the action of a group $G$ acting on $X$ is concentrated on a $G$-invariant subset $Y$ on which the $G$-action factors to that of an amenable group. The result is then applied to prove a conjecture of $S$. Wagon about finitely additive measures on spheres.
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