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Borel matrix. (English) Zbl 0837.28014
The author studies the Borel summation method. Let $$A= \{a_{n,k}, n,k\geq 1\}$$ be an infinite matrix with $$\sum^\infty_{k= 1} a_{n, k}= 1$$ $$(n\geq 1)$$ and $$\sup|a_n|_1< \infty$$. If $$X$$ is compact metrizable and $$\mu$$ a Borel probability measure, consider $$\widetilde S_n f= \sum^\infty_{k= 1} a_{n, k} f\circ T$$, where $$T$$ is the shift on $$(X^\infty, \mu^\infty)$$. If this converges a.e. to $$\int f d\mu$$ for all continuous $$f$$, $$A$$ is called Borel. Roughly speaking, a sequence $$a_n$$ in $$l_1$$ is a $$G(1)$$-set if there exist, for any $$\varepsilon> 0$$, sequences $$(b_n)$$ with $$\lim|b_n|_2= 0$$, $$\limsup|a_n- b_n|_1\leq \varepsilon$$ such that $$(b_n)$$ is a $$GC$$-set in $$l_2$$ (a concept defined by Dudley in terms of Gaussian processes).
For such $$A$$, the author proves the convergence $$\mu^\infty$$-a.e. of $$\widetilde S_n f$$ for $$f$$ in $$L_\infty(\mu^\infty)$$ of a probability space, and for other $$f$$’s “generated” from such special $$f$$’s. However, it is shown that the result cannot be extended to arbitrary dynamical systems. In the $$L_p$$-setting, necessary conditions are established using Bourgain’s entropy criterion.

##### MSC:
 28D05 Measure-preserving transformations 40G10 Abel, Borel and power series methods
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