Krylov–Bogolyubov theorem

The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after Russian-Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems.

Let X be a compact, metrizable topological space and F : X → X a continuous map.

Then F admits an invariant Borel probability measure.

be the transition probabilities for a time-homogeneous Markov semigroup on X, i.e. Theorem (Krylov–Bogolyubov).

for which the family of probability measures { Pt(x, ·) | t > 0 } is uniformly tight and the semigroup (Pt) satisfies the Feller property, then there exists at least one invariant measure for (Pt), i.e. a probability measure μ on X such that This article incorporates material from Krylov-Bogolubov theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.