Let be a measurable space and letf be a measurable function from X to itself. A measure μ on is said to be invariant underf if, for every measurable setA in Σ, In terms of the pushforward, this states that f∗ = μ. The collection of measures on X that are invariant under f is sometimes denoted Mf. The collection of ergodic measures, Ef, is a subset of Mf. Moreover, any convex combination of two invariant measures is also invariant, so Mf is a convex set; Ef consists precisely of the extreme points of Mf. In the case of a dynamical system, where is a measurable space as before, T is a monoid and φ : T × X → X is the flow map, a measure μ on is said to be an invariant measure if it is an invariant measure for each map φt : X → X. Explicitly, μ is invariant if and only if Put another way, μ is an invariant measure for a sequence of random variablest≥0 if, whenever the initial conditionZ0 is distributed according to μ, so is Zt 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
Consider the real lineR with its usual Borel σ-algebra; fixa ∈ R and consider the translation map Ta : R → R given by:
More generally, on n-dimensional Euclidean spaceRn with its usual Borel σ-algebra, n-dimensional Lebesgue measureλn is an invariant measure for any isometry of Euclidean space, i.e. a map T : Rn → Rn 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 S trivially has a decomposition into T-invariant components ' and '.
