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

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 $T^{\ast }f$ is finite a.e.