Let $(X,\mathcal A,\mu,T)$ be a probability-preserving system, meaning that $(X,\mathcal A,\mu)$ is a probability space, $T:X\to X$ is measurable, and $\mu(T^{-1}A)=\mu(A)$ for every $A\in\mathcal A$. If $T$ is invertible modulo $\mu$ and $\mathcal P$ is a countable measurable partition of $X$ such that
Here $h_\mu(T)$ denotes the supremum of $h_\mu(T,\mathcal Q)$ over all finite measurable partitions $\mathcal Q$ of $X$.
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For a non-invertible probability-preserving system, the same conclusion holds if $\mathcal P$ is a countable one-sided generator of finite entropy, meaning
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\begin{align*}
\sigma\left(\bigvee_{k=0}^{\infty}T^{-k}\mathcal P\right)=\mathcal A \quad \text{modulo } \mu.
\end{align*}