An ergodic Lebesgue differentiation theorem

Abstract

We show that if $(X, \mu, T)$ is a probability measure-preserving dynamical system, and $\mathscr{P}$ 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 $$\lim_{r \searrow 0, k \to \infty} \mu(B(x, r))^{-1} \int_{B(x, r)} \frac{1}{k} \sum_{j = 0}^{k - 1} f \circ T^j \mathrm{d} \mu$$ exists almost surely.