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Let $M_{\phi}$ denote the set of Borel probability measures invariant
under a topological action $\phi$ on a compact metrizable space $X$.
For a continuous function $f:X\to\R$, a measure $\mu\in\M_{\phi}$ is called
$f$-maximizing if $\int f\, d\mu = s u p\{\int f dm:m\in\M_{\phi}\}$. It
is shown that if $\mu$ is any ergodic measure in $\M_{\phi}$, then there
exists a continuous function whose unique maximizing measure is
$\mu$. More generally, if $\mathcal E$ is a non-empty collection of
ergodic measures which is weak$^*$ closed as a subset of $\M_{\phi}$, then
there exists a continuous function whose set of maximizing measures
is precisely the closed convex hull of $\mathcal E$. If moreover
$\phi$ has the property that its entropy map is upper
semi-continuous, then there exists a continuous function whose set
of equilibrium states is precisely the closed convex hull of
$\mathcal E$.