Invariant measure

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In mathematics, an invariant measure is a measure that is preserved by some function. Invariant measures are of great interest in the study of dynamical systems. The Krylov-Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.

[edit] Definition

Let (X, Σ) be a measurable space and let f be a measurable function from X to itself. A measure μ on (X, Σ) is said to be invariant under f if, for every measurable set A in Σ,

In terms of the push forward, this states that f_∗(μ) = μ.

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, φ), where (X, Σ) is a measurable space as before, T is a monoid and φ : T × X → X is the flow map, a measure μ on (X, Σ) is said to be an invariant measure if it is an invariant measure for each map φ_t : X → X. Explicity, μ is invariant if and only if

Put another way, μ is an invariant measure for a sequence of random variables (Z_t)_t≥0 (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition Z₀ is distributed according to μ, so is Z_t for any later time t.

[edit] Examples

• Consider the real line R with its usual Borel σ-algebra; fix a ∈ R and consider the translation map T_a : R → R given by:

T_a(x) = x + a.

Then one-dimensional Lebesgue measure λ is an invariant measure for T_a.

• More generally, on n-dimensional Euclidean space Rⁿ with its usual Borel σ-algebra, n-dimensional Lebesgue measure λⁿ is an invariant measure for any isometry of Euclidean space, i.e. a map T : Rⁿ → Rⁿ that can be written as

T(x) = Ax + b

for some n × n orthogonal matrix A ∈ O(n) and a vector b ∈ Rⁿ.