[proofplan] We use the variational principle to choose invariant probability measures whose measure-theoretic entropies approach the topological entropy. Compactness of $\mathcal{M}_T(X)$ gives a weak* convergent subsequence, and upper semicontinuity passes the entropy lower bound to the limit. The variational principle then gives the opposite inequality for the limiting measure, so the limit measure attains the topological entropy. [/proofplan]

[step:Choose invariant measures whose entropies approach the topological entropy]By the [variational principle for topological entropy](/theorems/6728) (citing a result not yet in the wiki: Variational Principle for Topological Entropy), \begin{align*} h_{\mathrm{top}}(T) = \sup_{\mu \in \mathcal{M}_T(X)} h_\mu(T). \end{align*} Since $h_{\mathrm{top}}(T) < \infty$, for each $j \in \mathbb{N}$ there exists a measure $\mu_j \in \mathcal{M}_T(X)$ such that \begin{align*} h_\mu_j(T) > h_{\mathrm{top}}(T) - \frac{1}{j}. \end{align*} Equivalently, \begin{align*} \lim_{j \to \infty} h_{\mu_j}(T) = h_{\mathrm{top}}(T). \end{align*}[/step]

[guided]The goal is to find an invariant measure whose entropy is exactly the supremal entropy. The variational principle for topological entropy says that the supremum of the measure-theoretic entropies over all invariant Borel probability measures is the topological entropy: \begin{align*} h_{\mathrm{top}}(T) = \sup_{\mu \in \mathcal{M}_T(X)} h_\mu(T). \end{align*} We are citing this as a prerequisite not yet resolved in the wiki: Variational Principle for Topological Entropy. Because $h_{\mathrm{top}}(T)$ is finite, the supremum is a finite real number. Therefore, by the defining property of the supremum, for every $j \in \mathbb{N}$ there exists $\mu_j \in \mathcal{M}_T(X)$ satisfying \begin{align*} h_{\mu_j}(T) > h_{\mathrm{top}}(T) - \frac{1}{j}. \end{align*} The variational principle also gives $h_{\mu_j}(T) \le h_{\mathrm{top}}(T)$ for every $j$, because $h_{\mathrm{top}}(T)$ is the supremum over all $\mathcal{M}_T(X)$. Hence \begin{align*} h_{\mathrm{top}}(T) - \frac{1}{j} < h_{\mu_j}(T) \le h_{\mathrm{top}}(T). \end{align*} The [squeeze theorem](/theorems/627) for real sequences gives \begin{align*} \lim_{j \to \infty} h_{\mu_j}(T) = h_{\mathrm{top}}(T). \end{align*}[/guided]

[step:Extract a weak* convergent subsequence in $\mathcal{M}_T(X)$] Since $X$ is compact metrizable and $T: X \to X$ is continuous, the space $\mathcal{M}_T(X)$ of $T$-invariant Borel probability measures is compact in the [weak* topology](/page/Weak*%20Topology). Hence there exist a subsequence $(\mu_{j_k})_{k \in \mathbb{N}}$ and a measure $\mu_* \in \mathcal{M}_T(X)$ such that \begin{align*} \mu_{j_k} \overset{*}{\rightharpoonup} \mu_* \end{align*} as $k \to \infty$. [/step]

[step:Use upper semicontinuity to pass entropy to the limit] Because the entropy map $\mu \mapsto h_\mu(T)$ is upper semicontinuous on $\mathcal{M}_T(X)$ and $\mu_{j_k} \overset{*}{\rightharpoonup} \mu_*$, we have \begin{align*} h_{\mu_*}(T) \ge \limsup_{k \to \infty} h_{\mu_{j_k}}(T). \end{align*} Since $(\mu_{j_k})$ is a subsequence of $(\mu_j)$ and $h_{\mu_j}(T) \to h_{\mathrm{top}}(T)$, it follows that \begin{align*} \limsup_{k \to \infty} h_{\mu_{j_k}}(T) = h_{\mathrm{top}}(T). \end{align*} Therefore \begin{align*} h_{\mu_*}(T) \ge h_{\mathrm{top}}(T). \end{align*} [/step]

[step:Apply the variational principle again to get equality] Since $\mu_* \in \mathcal{M}_T(X)$, the variational principle gives \begin{align*} h_{\mu_*}(T) \le \sup_{\mu \in \mathcal{M}_T(X)} h_\mu(T) = h_{\mathrm{top}}(T). \end{align*} Together with the opposite inequality obtained above, \begin{align*} h_{\mu_*}(T) \ge h_{\mathrm{top}}(T), \end{align*} we conclude that \begin{align*} h_{\mu_*}(T) = h_{\mathrm{top}}(T). \end{align*} Thus $\mu_*$ is a measure of maximal entropy for $(X,T)$. [/step]