 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 $H(\alpha \mid {\xi }_0^r)<\epsilon$. For $\gamma ={\alpha }_0^{n-1}$ and $\beta ={\xi }_0^{n+r-1}$, conditional subadditivity gives $H(\gamma \mid \beta )<n\epsilon$. Tail measurability and block conditional entropy given the infinite future give $H(\beta \mid \gamma )\ge (n+r)h$. Thus $H(\gamma )\le H(\beta )-(n+r)h+n\epsilon$. Divide by $n$, then let $n\to \infty$ and $\epsilon \downarrow 0$, proving $h_{\mu }(T,\alpha )=0$. The proof also works for noninvertible transformations.