= Finite partitions measurable in a partition tail have zero entropy rate
Let $\alpha$ be finite and measurable in $\mathcal T(\xi)$ for a finite <measurable partition> $\xi$, and put $h=h_\mu(T,\xi)$. Choose $r$ with $H(\alpha\mid\xi_0^r)<\varepsilon$. For $\gamma=\alpha_0^{n-1}$ and $\beta=\xi_0^{n+r-1}$, conditional subadditivity gives $H(\gamma\mid\beta)<n\varepsilon$. Tail measurability and <block conditional entropy given the infinite future> give $H(\beta\mid\gamma)\ge(n+r)h$. Thus $H(\gamma)\le H(\beta)-(n+r)h+n\varepsilon$. Divide by $n$, then let $n\to\infty$ and $\varepsilon\downarrow0$, proving $h_\mu(T,\alpha)=0$. The proof also works for noninvertible transformations.
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