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

A remarkable theorem of Besicovitch is that an integrable function f f on alttext="double-struck upper R squared"> R 2 encoding="application/x-tex">\mathbb {R}^2 strongly differentiable if its associated strong maximal alttext="upper M Subscript S Baseline f"> M S encoding="application/x-tex">M_S f finite a.e. We provide analogue Besicovitch’s result in the context ergodic theory provides a generalization Birkhoff’s Ergodic Theorem. In particular, we show measurable standard probability space and T"> T encoding="application/x-tex">T invertible measure-preserving transformation space, then averages with respect to converge only T Superscript asterisk ∗ encoding="application/x-tex">T^*f