mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 th ...

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 a p ...

. The theorems guarantee the existence of

invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, ...

s for certain "nice" maps defined on "nice" spaces and were named after

Russia Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...

Ukrainian Ukrainian may refer to: * Something of, from, or related to Ukraine * Something relating to Ukrainians, an East Slavic people from Eastern Europe * Something relating to demographics of Ukraine in terms of demography and population of Ukraine * So ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

s and

theoretical physicists The following is a partial list of notable theoretical physicists. Arranged by century of birth, then century of death, then year of birth, then year of death, then alphabetically by surname. For explanation of symbols, see Notes at end of this ar ...

Nikolay Krylov Nikolay Krylov may refer to: *Nikolay Krylov (marshal) (1903–1972), Soviet marshal *Nikolay Krylov (mathematician, born 1879) (1879–1955), Russian mathematician *Nikolay Krylov (mathematician, born 1941) (born 1941), Russian mathematician *Niko ...

and

Nikolay Bogolyubov Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretica ...

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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

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 , \infty) ...

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 points ...

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 value ...

. Then ''F'' admits an invariant

Borel Borel may refer to: People * Borel (author), 18th-century French playwright * Jacques Brunius, Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance * Émile Borel (1871 – 1956), a French mathematician known for his founding ...

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 gener ...

. That is, if Borel(''X'') denotes the

σ-algebra generated by the collection ''T'' of

open subsets In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...

of ''X'', then there exists a probability measure ''μ'' : Borel(''X'') →

, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...

such 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 In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...

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 t ...

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

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