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

2019/10/20 by Hagelstein, Paul, Herden, Daniel, Stokolos, Alexander
#37A30 #42B25 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.09054

Abstract

A remarkable theorem of Besicovitch is that an integrable function f on ℝ² 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.

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