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For a probability measure-preserving system, the hypothesis is completely positive entropy: every finite measurable partition with positive static entropy of a finite measurable partition has positive entropy rate of a measurable partition. Fix a finite partition $\xi$ and putAll sigma-algebras are interpreted modulo null sets. We will prove that every finite partition $\alpha$ measurable with respect to this tail sigma-algebra of a measurable partition has $h_{\mu }(T,\alpha )=0$. Applying this to a binary partition will force the required triviality. This proves the needed direction of the Tail characterization of the Pinsker sigma-algebra directly, including noninvertible transformations.

Write $h=h_{\mu }(T,\xi )$. The infinite-future entropy formula and the backwards chain rule for information entropy give, for every $L\ge 1$, the block conditional entropy given the infinite future identityEach term equals $h$ by invariance of the joint probabilities under a common pullback and continuity of conditional entropy under increasing finite future blocks. Invertibility is not needed for this identity.

Fix $\epsilon >0$. Since $\alpha$ is ${\mathcal{F}}_0$-measurable and finite, the martingale convergence theorem and continuity of finite-partition conditional entropy allow an $r\ge 0$ withTo see the continuity explicitly, for each partition atom $A$ of $\alpha$ the conditional probabilities $\mathbb{E}[1_A\mid \sigma ({\xi }_0^r)]$ tend to $1_A$ almost everywhere; apply dominated convergence theorem to the bounded continuous function $-t\log t$ on $[0,1]$ and sum over the finitely many partition atoms.

For $n\ge 1$ set $\gamma ={\alpha }_0^{n-1}$, $L=n+r$, and $\beta ={\xi }_0^{L-1}$. The conditional entropy of finite measurable partitions satisfiesThe second inequality uses conditioning reduces entropy, since $\beta$ refines each block $T^{-j}{\xi }_0^r$; the final equality uses measure preservation.

Moreover $\gamma$ is ${\mathcal{F}}_L$-measurable. For each $j\ge 0$, tail measurability gives $\alpha$ measurable with respect to ${\mathcal{F}}_L$, hence $T^{-j}\alpha$ measurable with respect to $T^{-j}{\mathcal{F}}_L={\mathcal{F}}_{L+j}\subseteq {\mathcal{F}}_L$. Thus conditioning reduces entropy and the displayed block identity implyUse the symmetric entropy identity $H(\gamma )=H(\beta )-H(\beta \mid \gamma )+H(\gamma \mid \beta )$ to obtainThe fraction on the right tends to zero: $r$ is fixed and $H({\xi }_0^{n+r-1})/(n+r)\to h$ by the definition of entropy rate of a measurable partition. Taking $n\to \infty$ and then $\epsilon \downarrow 0$ gives

Now let $A\in \mathcal{T}(\xi )$ and take $\alpha =\{A,X\setminus A\}$. If $0<\mu (A)<1$, its binary entropy is $-\mu (A)\log \mu (A)-(1-\mu (A))\log (1-\mu (A))>0$, while its entropy rate is zero, contradicting completely positive entropy. ThereforeSince $\xi$ was arbitrary, every finite-partition tail is trivial. The argument needs only that $\xi$ is finite and the measure is a probability; it does not assume a finite generator, finite total system entropy, or invertibility.

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.