  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.
