Every Ergodic Measure Is Uniquely Maximizing

Let Mφ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f : X → R, a measure μ ∈ Mφ is called f -maximizing if ∫ f dμ = sup{ ∫ f dm : m ∈Mφ}. It is shown that if μ is any ergodic measure in Mφ, then there exists a continuous function whose unique maximizing measure is μ. More generally, if E is a non-empty collection of ergodic measures which is weak∗ closed as a subset of Mφ, then there exists a continuous function whose set of maximizing measures is precisely the closed convex hull of E. If moreover φ 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 E.