## Almost everywhere convergence of weighted averages.(English)Zbl 0736.28008

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.

### MSC:

 28D05 Measure-preserving transformations
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### References:

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