= Infinite-future formula for partition entropy rate
{title2=$h_\mu(T,\xi)=H_\mu(\xi\mid\mathcal F_1)$}
For a finite <measurable partition> $\xi$, put $\mathcal F_1=\sigma(\bigvee_{j\ge1}T^{-j}\xi)$. Then $h_\mu(T,\xi)=H_\mu(\xi\mid\mathcal F_1)=\lim_{n\to\infty}H_\mu(\xi\mid\xi_1^n)$. The backward entropy chain rule writes $H_\mu(\xi_0^{N-1})$ as the sum of the first $N$ decreasing finite-future conditional entropies. Their averages have the same limit. Increasing conditioning fields converge by the <Martingale convergence theorem>.
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