Let $T:X\to X$ be a continuous map on a compact [metric space](/page/Metric%20Space), and let $\phi\in C(X)$. Then
\begin{align*}
P(T,\phi)=\sup_{\mu\in\mathcal M_T(X)}\left(h_\mu(T)+\int_X \phi\,d\mu\right),
\end{align*}
where $\mathcal M_T(X)$ is the set of $T$-invariant Borel probability measures on $X$.