  The classical theory of invariant means, which plays an important role in the
theory of paradoxical decompositions, is based upon what are usually termed
`pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean
inverse monoids which give rise to etale topological groupoids under
non-commutative Stone duality. We accordingly initiate the theory of invariant
means on arbitrary Boolean inverse monoids. Our main theorem is a
characterization of when a Boolean inverse monoid admits an invariant mean.
This generalizes the classical Tarski alternative proved, for example, by de la
Harpe and Skandalis, but using different methods.
