Almost everywhere convergence of convolution powers on compact Abelian groups (Q1951510)

Let \((X,\mathcal{B})\) be a measurable space and \(P: X\times\mathcal{B}\to [0,1]\) a transition probability with Markov operator \(Pf(x)= \int f(y) P(x, dy)\), for every bounded function \(f\). It is well known that the sequence \(\{P^n\}_{n\geq 1}\) has the strong sweeping out property if there exists a dense subset \(G_\delta\) of sets \(B\) of positive measure in the \(\mathcal{B}\) such that \(\limsup_n \mu^n*1_\beta= 1\) a.e. and \(\liminf_n \mu^n*1_\beta= 0\) a.e., for a strictly aperiodic probability measure \(\mu\). The aim of the present paper is to prove a necessary and sufficient condition for the strong sweeping out property for the convolution operator \(P_\mu\) defined by the relation \[ P_\mu f(x)= \mu* f(x)= \int_G f(x+ y)\,d\mu(y), \] for a strictly aperiodic probability measure \(\mu\) on a compact Abelian group \(G\), in terms of the Fourier-Stieltjes coefficients of \(\mu\). Furthermore, by extending the results to general compact Abelian groups \(G\) with Haar measure \(m\), the authors managed to obtain a dichotomy, namely, either \(\mu^n* f(x)\to \int f\,dm\) a.e. for every \(f\in L_p(G,m)\), \(p>1\), or \(\mu\) has the strong sweeping out property.