Invariant measure

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.

[1] 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.

be a measurable space and let

be a measurable function from

if, for every measurable set

In terms of the pushforward measure, this states that

( μ ) = μ .

The collection of measures (usually probability measures) on

is sometimes denoted

The collection of ergodic measures,

Moreover, any convex combination of two invariant measures is also invariant, so

is a convex set;

consists precisely of the extreme points of

In the case of a dynamical system

is a measurable space as before,

is the flow map, a measure

is said to be an invariant measure if it is an invariant measure for each map

Explicitly,

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

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

this being the largest eigenvalue as given by the Frobenius–Perron theorem.