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.
Categories: math.DS