Let $X$ be a compact [metrizable space](/page/Metrizable%20Space) and let $T: X \to X$ be continuous. Suppose that the topological entropy $h_{\mathrm{top}}(T)$ is finite and that the entropy map $\mu \mapsto h_\mu(T)$ is upper semicontinuous on the compact weak* space $\mathcal{M}_T(X)$ of $T$-invariant Borel probability measures. Then there exists $\mu_* \in \mathcal{M}_T(X)$ such that $h_{\mu_*}(T) = h_{\mathrm{top}}(T)$.