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

TL;DR

This paper extends Besicovitch's theorem on differentiability and maximal functions to ergodic theory, establishing a new criterion for almost everywhere convergence of ergodic averages based on maximal functions.

Contribution

It provides a novel ergodic-theoretic analogue of Besicovitch's differentiability theorem, linking maximal functions to convergence of ergodic averages.

Findings

Ergodic averages converge a.e. iff the ergodic maximal function is finite a.e.

Generalization of Birkhoff's Ergodic Theorem to include maximal function conditions.

Establishes a new criterion for pointwise convergence in ergodic theory.

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.