Kolmogorov extension theorem

All labels observed (6)

Label	Occurrences
Kolmogorov extension theorem canonical	4
Daniell–Kolmogorov theorem	1
Hahn–Kolmogorov theorem	1
Kolmogorov consistency conditions	1
Kolmogorov consistency theorem	1
Kolmogorov existence theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T3037540 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Kolmogorov extension theorem
Context triple: [Andrei Kolmogorov, notableWork, Kolmogorov extension theorem]

A. Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
B. Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
C. Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
D. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
E. Tonelli's theorem
Tonelli's theorem is a fundamental result in measure theory that justifies interchanging the order of integration for non-negative measurable functions in iterated Lebesgue integrals.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Kolmogorov extension theorem
Target entity description: The Kolmogorov extension theorem is a fundamental result in probability theory that guarantees the existence of a stochastic process with given consistent finite-dimensional distributions.

A. Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
B. Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
C. Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
D. Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
E. Tonelli's theorem
Tonelli's theorem is a fundamental result in measure theory that justifies interchanging the order of integration for non-negative measurable functions in iterated Lebesgue integrals.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	result in measure theory ⓘ theorem in probability theory ⓘ
appliesTo	collections of finite-dimensional distributions ⓘ product probability spaces ⓘ stochastic processes indexed by arbitrary index sets ⓘ
assumes	consistency under marginalization ⓘ each finite-dimensional distribution is a probability measure ⓘ measurability of coordinate projections ⓘ
concerns	cylinder sets ⓘ infinite product of measurable spaces ⓘ probability measures on path spaces ⓘ
conclusion	existence of a probability space and random variables with given joint laws ⓘ existence of a process whose finite-dimensional distributions match the given family ⓘ
coreConcept	extension of pre-measures to probability measures ⓘ finite-dimensional distributions determine a process under consistency ⓘ
field	measure theory ⓘ probability theory ⓘ stochastic processes ⓘ
formalizes	construction of laws of stochastic processes from finite-dimensional laws ⓘ construction of probability measures on function spaces ⓘ
guarantees	existence of a probability measure on an infinite product space ⓘ existence of a stochastic process with given finite-dimensional distributions ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	existence of a measure on the cylinder sigma-algebra ⓘ existence of a probability measure extending cylinder set measures ⓘ
namedAfter	Andrey Kolmogorov ⓘ linked to: Andrei Kolmogorov
relatedTo	Carathéodory extension theorem ⓘ linked to: Carathéodory’s extension theorem Daniell–Kolmogorov theorem ⓘ linked to: Kolmogorov extension theorem Kolmogorov consistency theorem ⓘ linked to: Kolmogorov extension theorem Kolmogorov continuity theorem ⓘ Kolmogorov existence theorem ⓘ linked to: Kolmogorov extension theorem projective limit of probability measures ⓘ
requires	Kolmogorov consistency conditions ⓘ linked to: Kolmogorov extension theorem consistency of finite-dimensional distributions ⓘ
usedFor	construction of Brownian motion ⓘ construction of Gaussian processes ⓘ construction of Markov processes ⓘ construction of random fields ⓘ construction of stationary processes ⓘ construction of stochastic processes ⓘ
usedIn	Bayesian nonparametrics ⓘ modern probability foundations ⓘ statistical mechanics models ⓘ theory of random functions ⓘ theory of random sequences ⓘ

How these facts were elicited

Referenced by (9)

Full triples — surface form annotated when it differs from this entity's canonical label.

Andrei Kolmogorov → notableWork → Kolmogorov extension theorem ⓘ

Carathéodory’s extension theorem → relatedTo → Kolmogorov extension theorem ⓘ

Carathéodory’s extension theorem → relatedTo → Hahn–Kolmogorov theorem ⓘ