An ergodic Lebesgue differentiation theorem

Abstract

We show that if $(X,\mu ,T)$ is a probability measure-preserving dynamical system, and $\mathcal{P}$ is a countable partition of $(X,\mu )$, then the limit \[ \lim_{n, k \to \infty} \mathbb{E} \left[ \frac{1}{k} \sum_{j = 0}^{k - 1} f \circ T^j \mid \bigvee_{i = 0}^{n - 1} T^{-i} \mathscr{P} \right] \] exists almost surely for all $f\in L^p(\mu ),p>1$. We prove this as a corollary of a geometric result: that if $(X,\mu )$ is a metric measure space on which the Hardy-Littlewood maximal inequality holds, then the limit \[\lim_{r \searrow 0, k \to \infty} \frac{1}{μ(B(x, r))} \int_{B(x, r)} \frac{1}{k} \sum_{j = 0}^{k - 1} f \circ T^j \mathrm{d} μ\] exists almost surely. In response to the first version of this article, J. Li has improved upon these results in a recent work, relaxing the assumption that $f\in L^p(\mu )$ for some $p>1$ to the assumption that $f\in L\log L(\mu )$, and relaxing some of the geometric assumptions we place on the underlying probability space. We will summarize his improvements.

arXiv commentUpdated in light of recent work by J. Li