A THEOREM OF BESICOVITCH AND A GENERALIZATION OF THE BIRKHOFF ERGODIC THEOREM

Paul Hagelstein, Daniel Herden, Alexander Stokolos

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

A remarkable theorem of Besicovitch is that an integrable function f on R2 is strongly differentiable if its associated strong maximal function MSf 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.

Original languageEnglish
Pages (from-to)52-59
Number of pages8
JournalProceedings of the American Mathematical Society, Series B
Volume8
DOIs
StatePublished - 2021

Keywords

  • Differentiation of integrals
  • maximal operators

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