An ergodic Lebesgue differentiation theorem

Abstract

We show that if $(X,\hat{I}¼,T)$ is a probability measure-preserving dynamical system, and $\mathcal{P}$ is a countable partition of $(X,\hat{I}¼)$, then the limit $$\underset{n,k\to \infty }{\lim }\mathbb{E}\left[\frac{1}{k}\sum _{j=0}^{k-1}f\circ T^j\mid \underset{i=0}{\overset{n-1}{\bigvee }}{T}^{-i}\mathcal{P}\right]$$ exists almost surely for all $f\in L^p(\hat{I}¼),p>1$. We prove this as a corollary of a geometric result: that if $(X,\hat{I}¼)$ is a metric measure space on which the Hardy-Littlewood maximal inequality holds, then the limit $$\underset{r\searrow 0,k\to \infty }{\lim }\hat{I}¼(B(x,r){)}^{-1}{\int }_{B(x,r)}\frac{1}{k}\sum _{j=0}^{k-1}f\circ T^j\mathrm{d}\hat{I}¼$$ exists almost surely.