An ergodic Lebesgue differentiation theorem
We show that if $(X, μ, T)$ is a probability measure-preserving dynamical system, and $\mathscr{P}$ is a countable partition of $(X, μ)$, 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 $f \in L^p(μ), p > 1$. We prove this as a corollary of a geometric result: that if $(X, μ)$ 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 $f \in L^p(μ)$ for some $p > 1$ to the assumption that $f \in L \log L(μ)$, and relaxing some of the geometric assumptions we place on the underlying probability space. We will summarize his improvements.