Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 108 / 3 / Solution

For a finite measurable partition $\xi$,Its conditional entropy of finite measurable partitions relative to $\eta$ isThe concavity of $-t\log t$, equivalently conditioning reduces entropy, gives

The atoms of $T^{-1}\xi$ are $T^{-1}A$ with the same measures as the atoms $A$ of $\xi$. Hence $\boxed{H_{\mu }(T^{-1}\xi )=H_{\mu }(\xi )}$.

SetThe chain rule and invariance give $a_{N+M}\le a_N+a_M$, so $(a_N)$ is a subadditive sequence. ThereforeThe Kolmogorov-Sinai entropy is $h_{\mu }(T)={\sup }_{\xi }{h}_{\mu }(T,\xi )$ over finite partitions.

Taking $F=\{0,\dots ,N-1\}$ immediately shows that the infimum over arbitrary finite $F$ is at most $h_{\mu }(T,\xi )$. For the reverse inequality, apply Shearer's inequality to translates of a fixed finite $F$ inside a long interval. Every interior coordinate is covered $|F|$ times, while only $O(\max F)$ boundary coordinates are lost. Subadditivity bounds the boundary contribution; division by the interval length and passage to the limit giveTaking the infimum proves