Let $(X,\mathcal B,\mu)$ be a standard probability space, let $T:X\to X$ be an ergodic measure-preserving transformation, and let $\mathcal P$ be a finite measurable partition. For $n\in\mathbb N$, write $\mathcal P_0^{n-1}:=\bigvee_{k=0}^{n-1}T^{-k}\mathcal P$ and $I_n^{\mathcal P}(x):=-\log\mu(\mathcal P_0^{n-1}(x))$. Then