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

Abstract

A remarkable theorem of Besicovitch is that an integrable function f f on R 2 \mathbb {R}^2 is strongly differentiable if its associated strong maximal function M S f 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 f is a measurable function on a standard probability space and T T is an invertible measure-preserving transformation on that space, then the ergodic averages of f f with respect to T T converge a.e. if and only if the associated ergodic maximal function T ∗ f T^*f is finite a.e.