Lebesgue decomposition theorem
E898510
The Lebesgue decomposition theorem is a fundamental result in measure theory that states any σ-finite measure can be uniquely decomposed into a part that is absolutely continuous with respect to another measure and a part that is singular to it.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lebesgue decomposition theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992085 — 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: Lebesgue decomposition theorem Context triple: [Henri Lebesgue, notableConcept, Lebesgue decomposition theorem]
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A.
Hahn decomposition theorem
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.
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B.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
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C.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
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D.
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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E.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
- 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: Lebesgue decomposition theorem Target entity description: The Lebesgue decomposition theorem is a fundamental result in measure theory that states any σ-finite measure can be uniquely decomposed into a part that is absolutely continuous with respect to another measure and a part that is singular to it.
-
A.
Hahn decomposition theorem
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.
-
B.
Lebesgue measure
Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
-
C.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
-
D.
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.
-
E.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in mathematical analysis
ⓘ
theorem in measure theory ⓘ |
| appliesTo |
finite measures
ⓘ
positive measures ⓘ σ-finite measures ⓘ |
| assumes | σ-algebra of measurable sets ⓘ |
| characterizes |
part of a measure that is absolutely continuous with respect to ν
ⓘ
part of a measure that is concentrated on a ν-null set ⓘ |
| classification | fundamental decomposition theorem for measures ⓘ |
| context |
integration with respect to different measures
ⓘ
modern measure-theoretic foundations of probability ⓘ |
| describes | decomposition of a measure relative to another measure ⓘ |
| field | measure theory ⓘ |
| formalStatement | Given σ-finite measures μ and ν on a measurable space, there exist unique measures μ_ac and μ_s such that μ = μ_ac + μ_s, μ_ac ≪ ν, and μ_s ⟂ ν NERFINISHED ⓘ |
| generalizes | Lebesgue decomposition of distribution functions ⓘ |
| guarantees |
uniqueness of the absolutely continuous component
ⓘ
uniqueness of the singular component ⓘ |
| hasConsequence |
every measure can be split into continuous and singular parts relative to a reference measure
ⓘ
structure theorem for measures on a measurable space ⓘ |
| holdsOn | a common measurable space for μ and ν ⓘ |
| implies | existence of a Radon–Nikodym derivative dμ_ac/dν ⓘ |
| involvesConcept |
Radon–Nikodym derivative
NERFINISHED
ⓘ
absolute continuity of measures ⓘ measure decomposition ⓘ mutual singularity of measures ⓘ singular measures ⓘ |
| isToolFor |
analyzing relationships between two measures
ⓘ
describing singular components of distributions ⓘ disintegrating probability measures ⓘ |
| namedAfter | Henri Lebesgue NERFINISHED ⓘ |
| relatedTo |
Jordan decomposition theorem
NERFINISHED
ⓘ
Lebesgue–Stieltjes measures NERFINISHED ⓘ Lebesgue’s decomposition of measures into discrete and continuous parts NERFINISHED ⓘ Radon–Nikodym theorem NERFINISHED ⓘ |
| requires | σ-finiteness of the reference measure ⓘ |
| statesThat |
any σ-finite measure μ can be decomposed into μ_ac + μ_s relative to another σ-finite measure ν
ⓘ
the decomposition μ = μ_ac + μ_s is unique ⓘ μ_ac is absolutely continuous with respect to ν ⓘ μ_s is singular with respect to ν ⓘ |
| symbolicForm | μ = μ_ac + μ_s with μ_ac ≪ ν and μ_s ⟂ ν ⓘ |
| usedIn |
ergodic theory
ⓘ
functional analysis ⓘ harmonic analysis ⓘ probability theory ⓘ spectral theory ⓘ stochastic processes ⓘ |
| usesNotation |
μ ≪ ν for absolute continuity
ⓘ
μ ⟂ ν for mutual singularity ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Lebesgue decomposition theorem Description of subject: The Lebesgue decomposition theorem is a fundamental result in measure theory that states any σ-finite measure can be uniquely decomposed into a part that is absolutely continuous with respect to another measure and a part that is singular to it.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.