Invariant measure

Let ${\displaystyle (X,\Sigma )}$ be a measurable space and let ${\displaystyle f:X\to X}$ be a measurable function from ${\displaystyle X}$ to itself. A measure ${\displaystyle \mu }$ on ${\displaystyle (X,\Sigma )}$ is said to be invariant under ${\displaystyle f}$ if, for every measurable set ${\displaystyle A}$ in ${\displaystyle \Sigma ,}$

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

In terms of the pushforward measure, this states that ${\displaystyle f_{*}(\mu )=\mu .}$

The collection of measures (usually probability measures) on ${\displaystyle X}$ that are invariant under ${\displaystyle f}$ is sometimes denoted ${\displaystyle M_{f}(X).}$ The collection of ergodic measures, ${\displaystyle E_{f}(X),}$ is a subset of ${\displaystyle M_{f}(X).}$ Moreover, any convex combination of two invariant measures is also invariant, so ${\displaystyle M_{f}(X)}$ is a convex set; ${\displaystyle E_{f}(X)}$ consists precisely of the extreme points of ${\displaystyle M_{f}(X).}$

In the case of a dynamical system ${\displaystyle (X,T,\varphi ),}$ where ${\displaystyle (X,\Sigma )}$ is a measurable space as before, ${\displaystyle T}$ is a monoid and ${\displaystyle \varphi :T\times X\to X}$ is the flow map, a measure ${\displaystyle \mu }$ on ${\displaystyle (X,\Sigma )}$ is said to be an invariant measure if it is an invariant measure for each map ${\displaystyle \varphi _{t}:X\to X.}$ Explicitly, ${\displaystyle \mu }$ is invariant if and only if

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

Put another way, ${\displaystyle \mu }$ is an invariant measure for a sequence of random variables ${\displaystyle \left(Z_{t}\right)_{t\geq 0}}$ (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition ${\displaystyle Z_{0}}$ is distributed according to ${\displaystyle \mu ,}$ so is ${\displaystyle Z_{t}}$ for any later time ${\displaystyle t.}$

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 ${\displaystyle 1,}$ 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 ${\displaystyle \mathbb {R} }$ with its usual Borel σ-algebra; fix ${\displaystyle a\in \mathbb {R} }$ and consider the translation map ${\displaystyle T_{a}:\mathbb {R} \to \mathbb {R} }$ given by:
  $${\displaystyle T_{a}(x)=x+a.}$$

  
  Then one-dimensional Lebesgue measure ${\displaystyle \lambda }$ is an invariant measure for ${\displaystyle T_{a}.}$
• More generally, on ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with its usual Borel σ-algebra, ${\displaystyle n}$-dimensional Lebesgue measure ${\displaystyle \lambda ^{n}}$ is an invariant measure for any isometry of Euclidean space, that is, a map ${\displaystyle T:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ that can be written as
  $${\displaystyle T(x)=Ax+b}$$

  
  for some ${\displaystyle n\times n}$ orthogonal matrix ${\displaystyle A\in O(n)}$ and a vector ${\displaystyle b\in \mathbb {R} ^{n}.}$
• 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 ${\displaystyle \mathbf {S} =\{A,B\}}$ and the identity map ${\displaystyle T=\operatorname {Id} }$ which leaves each point fixed. Then any probability measure ${\displaystyle \mu$ :\mathbf {S} \to \mathbb {R} } is invariant. Note that ${\displaystyle \mathbf {S} }$ trivially has a decomposition into ${\displaystyle T}$-invariant components ${\displaystyle \{A\}}$ and ${\displaystyle \{B\}.}$
• Area measure in the Euclidean plane is invariant under the special linear group ${\displaystyle \operatorname {SL} (2,\mathbb {R} )}$ of the ${\displaystyle 2\times 2}$ real matrices of determinant ${\displaystyle 1.}$
• Every locally compact group has a Haar measure that is invariant under the group action.

• Quasi-invariant measure

• John von Neumann (1999) Invariant measures, American Mathematical Society ISBN 978-0-8218-0912-9