Quasi-nilpotent sets in semigroups

Abstract:

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