Entropy rate of a measurable partition ($h_{\mu }(T,\xi )$)

The entropy rate of $\xi$ isThe numerator is a subadditive sequence, so the limit exists and equals the infimum of the displayed ratios.

The Kolmogorov-Sinai entropy is $h_{\mu }(T)={\sup }_{\xi }{h}_{\mu }(T,\xi )$ over finite measurable partitions.

The Pinsker sigma-algebra consists of the measurable sets $A$ whose binary partition $\{A,X\setminus A\}$ has entropy rate zero. It is a sigma-algebra and is the largest invariant factor with zero Kolmogorov-Sinai entropy.

A set belongs to the Pinsker sigma-algebra exactly when, modulo a null set, it belongs to $\mathcal{T}(\xi )$ for some finite measurable partition $\xi$. For a zero-entropy binary partition one may take that partition itself: zero conditional entropy makes its present atom measurable from every remote future.

For a measure-preserving action of $\mathbb{Z}$ and a finite partition $\xi$,The reverse inequality to the interval formula follows from Shearer's inequality applied to translates of $F$ along long intervals; the uncovered boundary has sublinear size.