Almost everywhere convergence of convolution powers on compact Abelian groups

It is well-known that a probability measure mu on the circle T satisfies parallel to mu(n) * f - integral fdm parallel to(p) -> 0 for every f is an element of L-p, every (some) p is an element of [1, infinity), if and only if vertical bar(mu) over cap (n)vertical bar < 1 for every non-zero n is an element of Z (mu, is strictly aperiodic). In this paper we study the a.e. convergence of mu(n) * f for every f is an element of L-p whenever p > 1. We prove a necessary and sufficient condition, in terms of the Fourier-Stieltjes coefficients of mu, for the strong sweeping out property (existence of a Borel set B with lim sup mu(n) * 1(B) = 1 a.e. and lim inf mu(n) * 1(B) = 0 a.e.). The results are extended to general compact Abelian groups G with Haar measure in, and as a corollary we obtain the dichotomy: for mu strictly aperiodic, either mu(n) * f -> integral f dm a.e. for every p > 1 and every f is an element of L-p (G, m), or mu, has the strong sweeping out property.