Hahn decomposition theorem

E475178

result in measure theory 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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (1)

Label	Occurrences
Hahn decomposition theorem canonical	2

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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
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 NERFINISHED ⓘ Lebesgue decomposition theorem NERFINISHED ⓘ Radon–Nikodym theorem NERFINISHED ⓘ
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 ⓘ

Referenced by (2)

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

Hans Hahn → notableWork → Hahn decomposition theorem ⓘ

measure theory → usesConcept → Hahn decomposition theorem ⓘ