 Solution

For a finite partition $\xi$, putThe system is K-mixing when every measurable $A$ becomes uniformly asymptotically independent of this remote future:Taking $\xi =\{B,X\setminus B\}$ makes $T^{-n}B\in {\mathcal{F}}_n(\xi )$, soThus K-mixing implies mixing.

The tail sigma-algebra of a measurable partition isBy the reverse martingale convergence theorem,in $L^1$. The uniform independence in the definition of K-mixing is equivalent to the limit being the constant $\mu (A)$ for every $A$. This holds exactly when every $\mathcal{T}(\xi )$-measurable set has measure zero or one. Hence the system is K-mixing if and only if every finite partition has trivial tail sigma-algebra.

Let $\mathcal{P}=\{A:h_{\mu }(T,{\xi }_A)=0\}$. Complements preserve the binary partition. If $A,B\in \mathcal{P}$, then ${\xi }_{A\cup B}$ is coarser than ${\xi }_A\vee {\xi }_B$, so subadditivity gives zero entropy rate. Thus $\mathcal{P}$ is an algebra. For $A={\bigcup }_{j\ge 1}{A}_j$, let $A^{(m)}={\bigcup }_{j\le m}{A}_j$. The entropy metric continuity boundtends to zero because $\mu (A△A^{(m)})\to 0$. Hence $A\in \mathcal{P}$, proving that $\mathcal{P}$ is a sigma-algebra: the Pinsker sigma-algebra.

If $A$ belongs modulo null sets to $\mathcal{T}(\xi )$ for a finite $\xi$, remote-future approximations make the entropy rate of ${\xi }_A$ zero. Conversely, if $h_{\mu }(T,{\xi }_A)=0$, the conditional-entropy formula for entropy rate givesThus $A$ is measurable modulo null sets from its strict future. Iterating this fact makes it measurable from every remote future, so $A\in \mathcal{T}({\xi }_A)$ modulo null sets. ThereforeThis is the Tail characterization of the Pinsker sigma-algebra.