In mathematics, the Krylov-Bogolyubov theorem (also known as the existence of invariant measures theorem) is a fundamental theorem in the study of dynamical systems. It guarantees the existence of invariant measures for "nice" maps on "nice" spaces. It is named after the Russian-Ukrainian mathemathicians and theoretical physicists Nikolay Mitrofanovich Krylov and Nikolay Bogolyubov
Statement of the theorem
Let (X, T) be a compact, metrizable topological space, and let F : X → X be a continuous map. Then F admits an invariant Borel probability measure. That is, if Borel(X) denotes the Borel σ-algebra generated by the collection T of open subsets of X, then there is a probability measure μ : Borel(X) → [0, 1] such that, for any subset A ∈ Borel(X),
- <math>\mu \left( F^{-1} (A) \right) = \mu (A).</math>
In terms of the push forward, this states that
- <math>F_{*} (\mu) = \mu.</math>
This article incorporates material from Krylov-Bogolubov theorem on PlanetMath, which is licensed under the GFDL.
