An ergodic Lebesgue differentiation theorem

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 $$\underset{n,k\to \mathrm{\infty }}{\lim }\mathbb{E}[\frac{1}{k}\sum _{j=0}^{k-1}f\circ T^j\mid \underset{i=0}{\overset{n-1}{\bigvee }}{T}^{-i}\mathcal{P}]$$ 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 $$\underset{r\searrow 0,k\to \mathrm{\infty }}{\lim }\mu (B(x,r){)}^{-1}{\int }_{B(x,r)}\frac{1}{k}\sum _{j=0}^{k-1}f\circ T^j\mathrm{d}\mu$$ exists almost surely.