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