Kolmogorov axioms
E320431
The Kolmogorov axioms are the standard mathematical foundation of probability theory, formalizing probabilities as measures on a sigma-algebra that satisfy non-negativity, normalization, and countable additivity.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kolmogorov axioms canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T3037534 — 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 axioms Context triple: [Andrei Kolmogorov, notableWork, Kolmogorov axioms]
-
A.
Shannon–Khinchin axioms
The Shannon–Khinchin axioms are a set of fundamental conditions that uniquely characterize Shannon entropy as the standard measure of information and uncertainty in probability theory and information theory.
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B.
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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C.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
-
D.
Foundations of Set Theory (with Andrey Kolmogorov)
"Foundations of Set Theory" is a classic 20th-century mathematical text co-authored by Pavel Alexandrov and Andrey Kolmogorov that systematically develops the basic concepts and axioms of set theory.
-
E.
Logical Foundations of Probability
Logical Foundations of Probability is a seminal philosophical work by Rudolf Carnap that develops a rigorous logical and formal account of probability and inductive reasoning.
- 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 axioms Target entity description: The Kolmogorov axioms are the standard mathematical foundation of probability theory, formalizing probabilities as measures on a sigma-algebra that satisfy non-negativity, normalization, and countable additivity.
-
A.
Shannon–Khinchin axioms
The Shannon–Khinchin axioms are a set of fundamental conditions that uniquely characterize Shannon entropy as the standard measure of information and uncertainty in probability theory and information theory.
-
B.
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.
-
C.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
-
D.
Foundations of Set Theory (with Andrey Kolmogorov)
"Foundations of Set Theory" is a classic 20th-century mathematical text co-authored by Pavel Alexandrov and Andrey Kolmogorov that systematically develops the basic concepts and axioms of set theory.
-
E.
Logical Foundations of Probability
Logical Foundations of Probability is a seminal philosophical work by Rudolf Carnap that develops a rigorous logical and formal account of probability and inductive reasoning.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
axiomatic system
ⓘ
foundation of probability theory ⓘ |
| acceptedAs | standard axioms of probability ⓘ |
| allows |
construction of product probability spaces
ⓘ
definition of expectation as Lebesgue integral ⓘ definition of random variable as measurable function ⓘ |
| appliesTo | sample space ⓘ |
| assumes |
finite additivity as a consequence of countable additivity
ⓘ
sigma-additivity ⓘ |
| axiom |
countable additivity of probability
ⓘ
non-negativity of probability ⓘ normalization of probability ⓘ |
| codomain | unit interval [0,1] ⓘ |
| compatibleWith |
Borel sigma-algebra
ⓘ
Lebesgue measure ⓘ |
| contrastWith |
frequentist interpretation of probability
ⓘ
subjective Bayesian interpretation of probability ⓘ |
| definesOn | event ⓘ |
| field |
measure theory
ⓘ
probability theory ⓘ |
| formalizes | probability measure ⓘ |
| formalizesAs | probability is a measure on a sigma-algebra ⓘ |
| generalizes | classical finite probability spaces ⓘ |
| hasDomain | sigma-algebra of subsets of sample space ⓘ |
| historicalPrecursor |
axioms of Andrey Markov
ⓘ
axioms of Émile Borel ⓘ |
| implies |
complement rule for probability
ⓘ
continuity from above ⓘ continuity from below ⓘ inclusion–exclusion principle ⓘ monotonicity of probability ⓘ probability of empty set equals 0 ⓘ subadditivity of probability ⓘ |
| influenced | measure-theoretic approach to probability ⓘ |
| languageOfOriginalFormulation | Russian ⓘ |
| mathematicalStructure | (Ω, F, P) probability space ⓘ |
| namedAfter |
Andrei Kolmogorov
ⓘ
surface form:
Andrey Kolmogorov
|
| numberOfCoreAxioms | 3 ⓘ |
| publicationYear | 1933 ⓘ |
| requires |
probability function is defined on all events in sigma-algebra
ⓘ
probability of any event is greater than or equal to 0 ⓘ probability of countable union of disjoint events equals sum of their probabilities ⓘ probability of the whole sample space equals 1 ⓘ |
| statedIn | Foundations of the Theory of Probability ⓘ |
| underlies |
mathematical statistics
ⓘ
modern probability theory ⓘ stochastic processes ⓘ |
| usesConcept |
measure space
ⓘ
sigma-algebra ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Kolmogorov axioms Description of subject: The Kolmogorov axioms are the standard mathematical foundation of probability theory, formalizing probabilities as measures on a sigma-algebra that satisfy non-negativity, normalization, and countable additivity.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Probability theory