Invariant measure

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,

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

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 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. Explicitly, μ is invariant if and only if

${\displaystyle \mu \left(\varphi _{t}^{-1}(A)\right)=\mu (A)\qquad \forall t\in T,A\in \Sigma .}$

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.

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.

Squeeze mapping leaves hyperbolic angle invariant as it moves a hyperbolic sector (purple) to one of the same area. Blue and green rectangles also keep the same area

• 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:

${\displaystyle T_{a}(x)=x+a.}$

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

${\displaystyle T(x)=Ax+b}$

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

• 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 ${\displaystyle {\boldsymbol {\rm {S}}}=\{A,B\}}$ and the identity map ${\displaystyle T={\rm {Id}}}$ which leaves each point fixed. Then any probability measure ${\displaystyle \mu :{\boldsymbol {\rm {S}}}\rightarrow {\boldsymbol {\rm {R}}}}$ is invariant. Note that S trivially has a decomposition into T-invariant components {A} and {B}.
• The measure of circular angles in degrees or radians is invariant under rotation. Similarly, the measure of hyperbolic angle is invariant under squeeze mapping.
• Area measure in the Euclidean plane is invariant under 2 × 2 real matrices with determinant 1, also known as the special linear group SL(2,R).
• Every locally compact group has a Haar measure that is invariant under the group action.

• Quasi-invariant measure

• Invariant measures, John Von Neumann, AMS Bookstore, 1999, ISBN 978-0-8218-0912-9