Block conditional entropy given the infinite future ($H_{\mu }({\xi }_0^{L-1}\mid {\mathcal{F}}_L)=L{h}_{\mu }(T,\xi )$)

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.