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.

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

- 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*.