Krylov–Bogolyubov theorem

In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. 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.

Theorem (Krylov–Bogolyubov). Let (X, T) be a compact, metrizable topological space and F : X → X 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 exists a probability measure μ : Borel(X) → [0, 1] such that for any subset A ∈ Borel(X),

In terms of the push forward, this states that

Let X be a Polish space and let ${\displaystyle P_{t},t\geq 0,}$ be the transition probabilities for a time-homogeneous Markov semigroup on X, i.e.

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