Let (X, \Sigma) be a
measurable space and let f : X \to X be a
measurable function from X to itself. A measure \mu on (X, \Sigma) is said to be
invariant under f if, for every measurable set A in \Sigma, \mu\left(f^{-1}(A)\right) = \mu(A). In terms of the
pushforward measure, this states that f_*(\mu) = \mu. The collection of measures (usually
probability measures) on X that are invariant under f is sometimes denoted M_f(X). The collection of
ergodic measures, E_f(X), is a subset of M_f(X). Moreover, any
convex combination of two invariant measures is also invariant, so M_f(X) is a
convex set; E_f(X) consists precisely of the extreme points of M_f(X). In the case of a
dynamical system (X, T, \varphi), where (X, \Sigma) is a measurable space as before, T is a
monoid and \varphi : T \times X \to X is the flow map, a measure \mu on (X, \Sigma) is said to be an
invariant measure if it is an invariant measure for each map \varphi_t : X \to X. Explicitly, \mu is invariant
if and only if \mu\left(\varphi_{t}^{-1}(A)\right) = \mu(A) \qquad \text{ for all } t \in T, A \in \Sigma. Put another way, \mu is an invariant measure for a sequence of
random variables \left(Z_t\right)_{t \geq 0} (perhaps a
Markov chain or the solution to a
stochastic differential equation) if, whenever the initial condition Z_0 is distributed according to \mu, so is Z_t for any later time t. When the dynamical system can be described by a
transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of 1, this being the largest eigenvalue as given by the
Frobenius–Perron theorem. ==Examples==