Almost everywhere convergence of convolution powers in \(L^ 1(X)\) (Q1320955)

Let \((X,\beta,m)\) be a probability measure space, \(\tau:X \to X\) an invertible measure preserving transformation and \(\mu\) a probability measure on the integers. The averages defined by the convolution powers of \(\mu\), \(\mu^ nf (x)= \sum^ \infty_{k=-\infty} \mu^ n(k) f(\tau^ kx)\), converge \(m\)-almost everywhere for all \(f \in L^ p\), \(1<p \leq \infty\), whenever \(\mu\) is a centered, aperiodic probability measure with finite second moment. We prove that these averages also converge in \(L^ 1(X)\) under similar conditions on \(\mu\). Subsequence results for measures which are not centered are given, as well as generalizations to measures on \(\mathbb{Z}^ d\) and \(\mathbb{R}^ d\).