Maximising measure

Definition

Let X be a topological space and let T : X → X be a continuous function. Let Inv(T) denote the set of all Borel probability measures on X that are invariant under T, i.e., for every Borel-measurable subset A of X, μ(T−1(A)) = μ(A). (Note that, by the Krylov-Bogolyubov theorem, if X is compact and metrizable, Inv(T) is non-empty.) Define, for continuous functions f : X → R, the maximum integral function β by

:\beta(f) := \sup \left. \left{ \int_{X} f , \mathrm{d} \nu \right| \nu \in \mathrm{Inv}(T) \right}.

A probability measure μ in Inv(T) is said to be a maximising measure for f if

:\int_{X} f , \mathrm{d} \mu = \beta(f).

Properties

• It can be shown that if X is a compact space, then Inv(T) is also compact with respect to the topology of weak convergence of measures. Hence, in this case, each continuous function f : X → R has at least one maximising measure.
• If T is a continuous map of a compact metric space X into itself and E is a topological vector space that is densely and continuously embedded in C(X; R), then the set of all f in E that have a unique maximising measure is equal to a countable intersection of open dense subsets of E.

References

• {{cite thesis
• {{cite journal | doi-access = free