measure theory

E400163

branch of mathematical analysis branch of mathematics

Measure theory is a branch of mathematical analysis that rigorously formalizes the concepts of length, area, volume, and integration for very general sets and functions.

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All labels observed (3)

Label	Occurrences
Measure Theory	2
measure theory canonical	2
real analysis	1

Statements (60)

Predicate	Object
instanceOf	branch of mathematical analysis ⓘ branch of mathematics ⓘ
appliesTo	complex-valued functions ⓘ functions on abstract spaces ⓘ real-valued functions ⓘ
developedIn	20th century ⓘ
fieldOfStudy	integration ⓘ measure ⓘ probability theory ⓘ real analysis ⓘ
formalizes	area ⓘ integration ⓘ length ⓘ volume ⓘ
foundationOf	ergodic theory ⓘ functional analysis ⓘ harmonic analysis ⓘ modern probability theory ⓘ
generalizes	Riemann integral ⓘ surface form: Riemann integration
hasKeyIdea	convergence theorems for integrals ⓘ countable additivity ⓘ integration with respect to a measure ⓘ measurability ⓘ sigma-additivity ⓘ
notableMeasure	Dirac delta function ⓘ surface form: Dirac measure Lebesgue measure ⓘ counting measure ⓘ probability measure ⓘ
usesConcept	Borel measure ⓘ Borel set ⓘ surface form: Borel sets Carathéodory’s extension theorem ⓘ surface form: Carathéodory extension theorem Fatou's lemma ⓘ surface form: Fatou lemma Fubini's theorem ⓘ surface form: Fubini theorem Hahn decomposition theorem ⓘ Jordan decomposition of measures ⓘ L^p space ⓘ Lebesgue integration ⓘ surface form: Lebesgue integral Lebesgue measure ⓘ Radon measure ⓘ Radon–Nikodym derivative ⓘ surface form: Radon–Nikodym theorem Tonelli's theorem ⓘ surface form: Tonelli theorem absolute continuity of measures ⓘ almost everywhere ⓘ completion of Borel sigma-algebra ⓘ completion of a measure ⓘ complex measure ⓘ dominated convergence theorem ⓘ indicator function ⓘ measurable function ⓘ measurable set ⓘ measure space ⓘ monotone convergence theorem ⓘ null set ⓘ outer measure ⓘ product measure ⓘ sigma-algebra ⓘ signed measure ⓘ simple function ⓘ singular measures ⓘ total variation of a measure ⓘ

Referenced by (5)

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

axiom of choice → usedIn → measure theory ⓘ

Weierstrass M-test → field → measure theory ⓘ

this entity surface form: real analysis

Joseph L. Doob → notableWork → measure theory ⓘ

this entity surface form: Measure Theory

Paul Halmos → notableWork → measure theory ⓘ

this entity surface form: Measure Theory

Edward Marczewski → fieldOfWork → measure theory ⓘ