Lebesgue decomposition theorem

E898510

result in mathematical analysis theorem in measure theory

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.

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All labels observed (1)

Label	Occurrences
Lebesgue decomposition theorem canonical	1

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 ⓘ

Referenced by (1)

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

Henri Lebesgue → notableConcept → Lebesgue decomposition theorem ⓘ