Infinite-future formula for partition entropy rate ($h_{\mu }(T,\xi )=H_{\mu }(\xi \mid {\mathcal{F}}_1)$)

For a finite measurable partition $\xi$, put ${\mathcal{F}}_1=\sigma ({\bigvee }_{j\ge 1}{T}^{-j}\xi )$. 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.

For ${\mathcal{F}}_L=\sigma ({\bigvee }_{j\ge L}{T}^{-j}\xi )$ and a finite measurable partition $\xi$, $H_{\mu }({\xi }_0^{L-1}\mid {\mathcal{F}}_L)=L{h}_{\mu }(T,\xi )$. The backward conditional entropy chain rule expresses the left side as ${\sum }_{j=0}^{L-1}{H}_{\mu }(T^{-j}\xi \mid {\mathcal{F}}_{j+1})$. Each term equals $h_{\mu }(T,\xi )$ by measure preservation and the infinite-future formula for partition entropy rate. No invertibility is needed.