Almost everywhere convergence of weighted averages
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 \(f\in L^ p(m)\), \(1<p\leq\infty\). In particular, if \(\lim_{n\to\infty}\sum^{\infty}_{k=-\infty}|\mu_ n(k)-\mu_ n(k-1)|=0\), then there exists a subsequence \((\mu_{n_ m})\) such that for all systems \((X,\beta,m,\tau)\) and \(f\in L^ p(m), 1<p\leq\infty\), \(\lim_{n\to\infty}\mu_{n_ m}f(x)\) exists a.e. If \(\mu_ n=\mu^ n\) for some fixed strictly aperiodic finitely supported measure \(\mu\), then \((n_ m)\) can be any subsequence with \(n_{m+1}>n^{\beta}_ m\) for some \(\beta>1\). Related theorems, examples, and generalizations to group actions are also described.
invertible measure-preserving transformation
scientific article; zbMATH DE number 4215991
scientific article