| Abstract: |
| A classical result of Besicovitch is that a measurable function $f$ on $\mathbb{R}^2$ is strongly differentiable if $M_S f < \infty$ a.e. In this talk we consider analogues of this result in ergodic theory, and in particular we provide the following generalization of Birkhoff`s Ergodic Theorem:
Suppose $T$ is an invertible measure-preserving transformation on the probability space $(X, \mathcal{B}, \mu)$ and $f$ is a $\mu$-measurable (not necessarily integrable, and not necessarily nonnegative) function on $X$. If the ergodic maximal function $T^\ast f(x)$ is finite $\mu$-a.e., where $T^\ast f$ is defined by
$$T^\ast f(x) = \sup_{n \geq 1} \frac{1}{n}\left|\sum_{k=0}^{n-1} f(T^k x)\right|,$$
then the limit
$$\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} f(T^kx)$$
exists $\mu$-a.e.
This research is joint with Daniel Herden and Alex Stokolos. |
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