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

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

s within the theory of

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

. The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after

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n- Ukrainian

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s and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems. Zbl. 16.86.

Formulation of the theorems

Invariant measures for a single map

Theorem (Krylov–Bogolyubov). Let (''X'', ''T'') be a compact,

metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inf ...

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

and ''F'' : ''X'' → ''X'' a

continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...

. Then ''F'' admits an invariant Borel

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...

. 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'') → , 1such that for any subset ''A'' ∈ Borel(''X''), :

\mu \left( F^ (A) \right) = \mu (A).

In terms of the push forward, this states that :

F_ (\mu) = \mu.

Invariant measures for a Markov process

Let ''X'' be a Polish space and let

P_t, t\ge 0,

be the transition probabilities for a time-homogeneous

Markov Markov ( Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include: Academics * Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology at ...

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on ''X'', i.e. :

\Pr X_ = x = P_ (x, A).

Theorem (Krylov–Bogolyubov). If there exists a point

x\in X

for which the family of probability measures is uniformly tight and the semigroup (''P''_''t'') satisfies the Feller property, then there exists at least one invariant measure for (''P''_''t''), i.e. a probability measure ''μ'' on ''X'' such that :

(P_)_ (\mu) = \mu \mbox t > 0.

Notes

{{DEFAULTSORT:Krylov-Bogolyubov theorem Ergodic theory Theorems in dynamical systems Probability theorems Random dynamical systems Theorems in measure theory

Formulation of the theorems

Invariant measures for a single map

Invariant measures for a Markov process

See also

Notes