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, \mathcal{A}) be a measurable space and let f : X \to X be a measurable function. A measure μ on (X, \mathcal{A}) is said to be invariant under f if, for every measurable set A \in \mathcal{A},

\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).

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