A theorem of Besicovitch and a generalization of the Birkhoff Ergodic Theorem

Paul Hagelstein; Daniel Herden; Alexander Stokolos

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arXiv · 1910.09054

A theorem of Besicovitch and a generalization of the Birkhoff Ergodic Theorem

Paul Hagelstein·Daniel Herden·Alexander Stokolos

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Abstract

A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result in the context of ergodic theory that provides a generalization of Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable function on a standard probability space and $T$ is an invertible measure-preserving transformation on that space, then the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated ergodic maximal function $T^*f$ is finite a.e.

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BibTeX RIS

Paul Hagelstein, Daniel Herden, Alexander Stokolos. 2019-10-20. A theorem of Besicovitch and a generalization of the Birkhoff Ergodic Theorem. https://arxiv.org/abs/1910.09054

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