In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.

Definition

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, In terms of the pushforward measure, this states that 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 where (X, \Sigma) is a measurable space as before, T is a monoid and 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 Explicitly, \mu is invariant if and only if Put another way, \mu is an invariant measure for a sequence of random variables (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