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,

\mu \left( f^{-1} (A) \right) = \mu (A).

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 Mf(X). The collection of ergodic measures, Ef(X), is a subset of Mf(X). Moreover, any convex combination of two invariant measures is also invariant, so Mf(X) is a convex set; Ef(X) consists precisely of the extreme points of Mf(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 φt : X → X. Explicity, μ is invariant if and only if

\mu \left( \varphi_{t}^{-1} (A) \right) = \mu (A) \forall t \in T, A \in \Sigma.

Put another way, μ is an invariant measure for a sequence of random variables (Zt)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 Zt 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 Ta : R → R given by:
Ta(x) = x + a.
Then one-dimensional Lebesgue measure λ is an invariant measure for Ta.
  • 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.