Given a sequence \((\mu_ n)\) of probability measures on \(Z\), and an invertible measure-preserving transformation \(\tau\) of a probability space \((X,\beta,m)\), the averages \(\mu_ nf(x)=\sum^{\infty}_{k=- \infty}\mu_ n(k)f(\tau^ kx)\) are bounded operators on \(L^ p(m)\), \(1\leq p\leq\infty\). Sufficient conditions are given in terms of the Fourier transforms \((\hat\mu_ n(\gamma))\) on \(T=\{\gamma:|\gamma|=1\}\) for the a.e. convergence of \(\mu_ nf(x)\) for all \(f\in L^ p(m)\), \(1n^{\beta}_ m\) for some \(\beta>1\). Related theorems, examples, and generalizations to group actions are also described.