Mean ergodic theorems for power bounded measures

Abstract

Let G be a locally compact abelian group and let M(G) be the convolution measure algebra of G. A measure μ∈M(G) is said to be power bounded if supn≥0⁡‖μn‖1<∞, where μn denotes nth convolution power of μ. We show that if μ∈M(G) is power bounded and A=[an,k]n,k=0∞ is a strongly regular matrix, then the limit limn→∞⁡∑k=0∞an,kμk exists in the weak⁎ topology of M(G) and is equal to the idempotent measure θ, where θˆ=1intFμ. Here, θˆ is the Fourier-Stieltjes transform of θ, Fμ:={γ∈Γ:μˆ(γ)=1}, and 1intFμ is the characteristic function of intFμ. Some applications are also given.