The collection of measures (usually
probability measures) on that are invariant under is sometimes denoted The collection of
ergodic measures, is a subset of 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 where is a measurable space as before, is a
monoid and is the flow map, a measure on is said to be an invariant measure if it is an invariant measure for each map Explicitly, is invariant
if and only if
Put another way, 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 is distributed according to so is for any later time
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.
Examples
Consider the
real line with its usual
Borel σ-algebra; fix and consider the translation map given by:
Then one-dimensional
Lebesgue measure is an invariant measure for
More generally, on -dimensional
Euclidean space with its usual Borel σ-algebra, -dimensional Lebesgue measure is an invariant measure for any
isometry of Euclidean space, that is, a map that can be written as
The invariant measure in the first example is unique up to trivial renormalization with a constant factor. This does not have to be necessarily the case: Consider a set consisting of just two points and the identity map which leaves each point fixed. Then any probability measure is invariant. Note that trivially has a decomposition into -invariant components and