Quasi-nilpotent sets in semigroups

Abstract

In a compact semigroup S S with zero 0 0 , a subset A A of S S is called quasi-nilpotent if the closed semigroup generated by A A contains 0 0 . A probability measure μ \mu on S S is called nilpotent if the sequence ( μ n ) ({\mu ^n}) converges to the Dirac measure at 0 0 . It is shown that a probability measure is nilpotent if and only if its support is quasi-nilpotent. Consequently, the set of all nilpotent measures on S S is convex and everywhere dense in the set of all probability measures on S S and the union of their supports is S S .