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# Almost Everywhere Convergence of Convolution Measures

Published:2011-06-24
Printed: Dec 2012
• Karin Reinhold,
Department of Mathematics, University at Albany, SUNY, Albany, NY 12222 USA
• Anna K. Savvopoulou,
Department of Mathematical Sciences, Indiana University in South Bend, South Bend, IN, 46545 USA
• Christopher M. Wedrychowicz,
Department of Mathematical Sciences, Indiana University in South Bend, South Bend, IN, 46545 USA
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## Abstract

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.
 MSC Classifications: 28D - unknown classification 28D

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