= Solution
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 put
$$
\mathcal F_N=\sigma\left(\bigvee_{j\ge N}T^{-j}\xi\right),\qquad\mathcal T(\xi)=\bigcap_{N\ge0}\mathcal F_N.
$$
All <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 $L\ge1$, the <block conditional entropy given the infinite future> identity
$$
\boxed{H_\mu(\xi_0^{L-1}\mid\mathcal F_L)=\sum_{j=0}^{L-1}H_\mu(T^{-j}\xi\mid\mathcal F_{j+1})=Lh}.
$$
Each 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 $\varepsilon>0$. Since $\alpha$ is $\mathcal F_0$-measurable and finite, the <Martingale convergence theorem> and continuity of finite-partition <conditional entropy> allow an $r\ge0$ with
$$
H_\mu(\alpha\mid\xi_0^r)<\varepsilon.
$$
To see the continuity explicitly, for each <partition atom> $A$ of $\alpha$ the conditional probabilities $\mathbb E[\mathbf1_A\mid\sigma(\xi_0^r)]$ tend to $\mathbf1_A$ 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 $n\ge1$ set $\gamma=\alpha_0^{n-1}$, $L=n+r$, and $\beta=\xi_0^{L-1}$. The <conditional entropy of finite measurable partitions> satisfies
$$
H(\gamma\mid\beta)\le\sum_{j=0}^{n-1}H(T^{-j}\alpha\mid\beta)\le\sum_{j=0}^{n-1}H(T^{-j}\alpha\mid T^{-j}\xi_0^r)=nH(\alpha\mid\xi_0^r)<n\varepsilon.
$$
The 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 $j\ge0$, 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 imply
$$
H(\beta\mid\gamma)\ge H(\beta\mid\mathcal F_L)=Lh.
$$
Use the symmetric entropy identity $H(\gamma)=H(\beta)-H(\beta\mid\gamma)+H(\gamma\mid\beta)$ to obtain
$$
0\le\frac1nH(\alpha_0^{n-1})\le\frac{H(\xi_0^{n+r-1})-(n+r)h}{n}+\varepsilon.
$$
The 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 $\varepsilon\downarrow0$ gives
$$
\boxed{h_\mu(T,\alpha)=0\quad\text{for every finite }\mathcal T(\xi)\text{-measurable partition }\alpha}.
$$
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>. Therefore
$$
\boxed{\mu(A)\in\{0,1\}\quad\text{for every }A\in\mathcal T(\xi)}.
$$
Since $\xi$ 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.
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