Nilpotent measures on compact semigroups

Abstract

LetSbe a compact semigroup andP(S) the set of probability measures onS. SupposeP(S) has zero θ and define a measure μ εP(S) nilpotent if μn→ θ. It is shown that any measure with support containing that of θ is nilpotent, and the set of nilpotent measures is convex and dense inP(S). A measure μ is called mean-nilpotent if (μ + μ2+ … + μn)/n→ θ, and can be characterized in terms of its support.