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 language | English |
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Pages (from-to) | 52-59 |
Number of pages | 8 |
Journal | Proceedings of the American Mathematical Society, Series B |
Volume | 8 |
DOIs | |
State | Published - 2021 |
Keywords
- Differentiation of integrals
- maximal operators