Invariant measure

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

Definition[edit]

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 (XTφ), 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. Explicitly, μ is invariant if and only if

\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 (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.

Examples[edit]

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 aR and consider the translation map Ta : RR given by:
T_{a} (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 : RnRn that can be written as
T(x) = A x + b
for some n × n orthogonal matrix A ∈ O(n) and a vector bRn.
  • 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  \boldsymbol{\rm S}=\{A,B\} and the identity map T={\rm Id} which leaves each point fixed. Then any probability measure \mu : \boldsymbol{\rm S} \rightarrow \boldsymbol{\rm R} is invariant. Note that S trivially has a decomposition into T-invariant components {A} and {B}.

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