Almost Everywhere Convergence of Convolution Measures

http://dx.doi.org/10.4153/CMB-2011-124-3
Canad. Math. Bull. 55(2012), 830-841

  Published:2011-06-24
 Printed: Dec 2012

Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $(X,\mathcal{B},m)$ a probability space and $\tau$ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in $\textrm{L}^1(X)$ of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures $\{\nu_i\}$ defined on $\mathbb{Z}$. We then exhibit cases of such averages where convergence fails.