Maximising measure

In mathematics — specifically, in ergodic theory — a maximising measure is a particular kind of probability measure. Informally, a probability measure "μ" is a maximising measure for some function "f" if the integral of "f" with respect to "μ" is “as big as it can be”. The theory of maximising measures is relatively young and quite little is known about their general structure and properties.
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
A probability measure "μ" in Inv("T") is said to be a maximising measure for "f" if