Let $(X,\mathcal{F},\mu)$ be a probability space, and let $T:X\to X$ be a measurable measure-preserving map. Let $\mathcal{P}$ and $\mathcal{Q}$ be finite measurable partitions of $X$. Suppose that $\mathcal{Q}$ refines $\mathcal{P}$, meaning that for every atom $Q\in\mathcal{Q}$ there exists an atom $P\in\mathcal{P}$ such that $Q\subset P$. Then

\begin{align*} h_\mu(T,\mathcal{P})\leq h_\mu(T,\mathcal{Q}). \end{align*}