Hahn decomposition theorem
E475178
The Hahn decomposition theorem is a fundamental result in measure theory that states any signed measure space can be partitioned into a positive set and a negative set on which the measure is respectively nonnegative and nonpositive.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hahn decomposition theorem canonical | 2 |
| Jordan decomposition theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4857418 — 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: Hahn decomposition theorem Context triple: [Hans Hahn, notableWork, Hahn decomposition theorem]
-
A.
Doob–Meyer decomposition
The Doob–Meyer decomposition is a fundamental result in stochastic process theory that uniquely expresses a submartingale as the sum of a martingale and a predictable, increasing process.
-
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.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
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D.
Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
-
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: Hahn decomposition theorem Target entity description: The Hahn decomposition theorem is a fundamental result in measure theory that states any signed measure space can be partitioned into a positive set and a negative set on which the measure is respectively nonnegative and nonpositive.
-
A.
Doob–Meyer decomposition
The Doob–Meyer decomposition is a fundamental result in stochastic process theory that uniquely expresses a submartingale as the sum of a martingale and a predictable, increasing process.
-
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.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
-
D.
Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
-
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 (39)
| Predicate | Object |
|---|---|
| instanceOf |
result in measure theory
ⓘ
theorem ⓘ |
| appearsIn |
textbooks on probability theory
ⓘ
textbooks on real analysis ⓘ |
| appliesTo |
signed measure
ⓘ
signed measure space ⓘ |
| assumes |
a signed measure defined on a σ-algebra
ⓘ
a σ-algebra of measurable sets ⓘ |
| conclusion | there exists a measurable positive set P and a measurable negative set N with union equal to the whole space up to a null set ⓘ |
| ensures |
existence of maximal negative sets for a signed measure
ⓘ
existence of maximal positive sets for a signed measure ⓘ |
| field | measure theory ⓘ |
| hasConsequence |
Jordan decomposition theorem
ⓘ
linked to:
Lebesgue decomposition theorem
|
| holdsFor |
finite signed measures
ⓘ
general signed measures on measurable spaces ⓘ σ-finite signed measures ⓘ |
| implies |
existence of a negative set for a signed measure
ⓘ
existence of a positive set for a signed measure ⓘ the underlying space is the union of a positive set and a negative set up to a null set ⓘ |
| mathematicalDomain |
functional analysis
ⓘ
probability theory ⓘ real analysis ⓘ |
| namedAfter | Hans Hahn ⓘ |
| property |
the negative set is not unique but is unique up to a null set
ⓘ
the positive set is not unique but is unique up to a null set ⓘ |
| relatedTo |
Jordan decomposition theorem
ⓘ
linked to:
Lebesgue decomposition theorem
Lebesgue decomposition theorem ⓘ Radon–Nikodym theorem ⓘ
linked to:
Radon–Nikodym derivative
|
| statesThat |
every signed measure space admits a decomposition into a positive set and a negative set
ⓘ
the positive set and negative set form a partition of the underlying space up to a null set ⓘ there exists a measurable set N such that the signed measure is nonpositive on all measurable subsets of N ⓘ there exists a measurable set P such that the signed measure is nonnegative on all measurable subsets of P ⓘ |
| topicOf | graduate-level measure theory courses ⓘ |
| usedFor |
analysis of signed measures
ⓘ
construction of Jordan decomposition ⓘ decomposition of signed measures into positive and negative parts ⓘ |
| usedIn |
functional analysis
ⓘ
integration theory ⓘ probability theory with signed or finite signed measures ⓘ |
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: Hahn decomposition theorem Description of subject: The Hahn decomposition theorem is a fundamental result in measure theory that states any signed measure space can be partitioned into a positive set and a negative set on which the measure is respectively nonnegative and nonpositive.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hahn decomposition theorem