Kolmogorov zero–one law
E320434
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kolmogorov zero–one law canonical | 2 |
| zero–one law | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3037539 — 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 zero–one law Context triple: [Andrei Kolmogorov, notableWork, Kolmogorov zero–one law]
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A.
Kakutani’s random ergodic theorem
Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
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B.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
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C.
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.
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D.
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.
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E.
Modern Probability Theory and Its Applications
"Modern Probability Theory and Its Applications" is a foundational textbook by Emanuel Parzen that systematically develops modern probability theory and demonstrates its use in a wide range of statistical and applied contexts.
- 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 zero–one law Target entity description: The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
-
A.
Kakutani’s random ergodic theorem
Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
-
B.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
C.
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.
-
D.
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.
-
E.
Modern Probability Theory and Its Applications
"Modern Probability Theory and Its Applications" is a foundational textbook by Emanuel Parzen that systematically develops modern probability theory and demonstrates its use in a wide range of statistical and applied contexts.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
probability theorem
ⓘ
result in probability theory ⓘ |
| appliesTo |
independent random variables
ⓘ
sequences of independent random variables ⓘ |
| assumes | mutual independence of the random variables ⓘ |
| characterizes | triviality of the tail sigma-algebra for independent sequences ⓘ |
| concerns |
events invariant under finite modifications of coordinates
ⓘ
tail events ⓘ |
| conclusionType | zero–one valued probabilities ⓘ |
| contrastsWith | events depending on finitely many coordinates ⓘ |
| ensures | no nontrivial tail events with intermediate probability ⓘ |
| field |
measure theory
ⓘ
probability theory ⓘ |
| formalSetting |
probability space
ⓘ
product probability space ⓘ |
| hasConsequence | tail sigma-algebra is almost surely trivial ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| holdsFor | countable sequences of independent random variables ⓘ |
| implies | tail events are almost sure or almost impossible ⓘ |
| involves | events measurable with respect to the tail sigma-algebra ⓘ |
| isPartOf | classical probability theory ⓘ |
| isTaughtIn |
advanced measure-theoretic probability textbooks
ⓘ
graduate probability courses ⓘ |
| mathematicalDomain |
infinite product measures
ⓘ
probability on product spaces ⓘ |
| namedAfter |
Andrey Kolmogorov
ⓘ
linked to:
Andrei Kolmogorov
|
| relatedTo |
Borel–Cantelli lemmas
ⓘ
Hewitt–Savage zero–one law ⓘ ergodic theorems ⓘ law of large numbers ⓘ |
| requires | countable additivity of the probability measure ⓘ |
| statesThat | every tail event of a sequence of independent random variables has probability 0 or 1 ⓘ |
| typeOfResult |
zero–one law
ⓘ
linked to:
Kolmogorov zero–one law
|
| typicalProofUses |
independence and invariance arguments
ⓘ
properties of conditional expectation ⓘ |
| usedIn |
analysis of convergence of random variables
ⓘ
analysis of random series ⓘ ergodic theory ⓘ probabilistic number theory ⓘ theory of stochastic processes ⓘ |
| usesConcept |
almost sure events
ⓘ
independence of sigma-algebras ⓘ probability measure ⓘ sigma-algebra ⓘ tail sigma-algebra ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Kolmogorov zero–one law Description of subject: The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Kolmogorov zero–one law