Lebesgue integration

E59633

integration theory mathematical concept measure-theoretic construction

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.

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All labels observed (3)

Label	Occurrences
Lebesgue integral	11
Lebesgue integration canonical	5
Lebesgue integral on the real line	1

Statements (50)

Predicate	Object
instanceOf	integration theory ⓘ mathematical concept ⓘ measure-theoretic construction ⓘ
advantageOver	Riemann integration in dealing with sets of measure zero ⓘ Riemann integration in dealing with unbounded functions ⓘ Riemann integration in handling limits of sequences of functions ⓘ
allows	integration of a wider class of functions than Riemann integration ⓘ
appliesTo	functions on general measure spaces ⓘ functions on ℝ ⓘ
approach	partition of the range of the function ⓘ
basedOn	measure ⓘ sigma-algebra ⓘ
characterizedBy	Fatou's lemma ⓘ construction via simple functions ⓘ countable additivity over disjoint measurable sets ⓘ dominated convergence theorem ⓘ integration with respect to a measure ⓘ linearity of the integral ⓘ monotone convergence theorem ⓘ
contrastWith	Riemann integration's partition-of-domain approach ⓘ
domain	functions defined almost everywhere ⓘ measurable functions ⓘ
ensures	completeness of L^p spaces ⓘ existence of integrals for many pointwise limits of integrable functions ⓘ
field	functional analysis ⓘ measure theory ⓘ real analysis ⓘ
generalizes	Riemann integral ⓘ surface form: Riemann integration
historicalPeriod	early 20th century ⓘ
introducedBy	Henri Lebesgue ⓘ
namedAfter	Henri Lebesgue ⓘ
relatedTo	Fubini's theorem ⓘ Radon–Nikodym derivative ⓘ surface form: Radon–Nikodym theorem Tonelli's theorem ⓘ change of variables formula in measure theory ⓘ
supports	Fourier analysis on L^2 ⓘ Hilbert space L^2 ⓘ L^p spaces ⓘ
underlies	expectation of random variables ⓘ modern probability theory ⓘ stochastic processes ⓘ
usedIn	ergodic theory ⓘ functional analysis ⓘ harmonic analysis ⓘ partial differential equations ⓘ
usesConcept	Lebesgue measure ⓘ almost everywhere equality ⓘ measurable functions ⓘ measurable sets ⓘ null sets ⓘ