= Block conditional entropy given the infinite future
{title2=$H_\mu(\xi_0^{L-1}\mid\mathcal F_L)=Lh_\mu(T,\xi)$}
For $\mathcal F_L=\sigma(\bigvee_{j\ge L}T^{-j}\xi)$ and a finite <measurable partition> $\xi$, $H_\mu(\xi_0^{L-1}\mid\mathcal F_L)=Lh_\mu(T,\xi)$. 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.
Back to article page
