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.

Aliases (1)

Lebesgue integral ×4

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 →

Referenced by (6)

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

Radon–Nikodym derivative → relatedTo → Lebesgue integration →

this entity surface form: "Lebesgue integral"

Riemann sums → relatedTo → Lebesgue integration →

this entity surface form: "Lebesgue integral"

Riemann–Lebesgue lemma → relatedTo → Lebesgue integration →

this entity surface form: "Lebesgue integral"

Gaussian integral → type → Lebesgue integration →

this entity surface form: "Lebesgue integral"

Hölder inequality → usedIn → Lebesgue integration →

Itô calculus → uses → Lebesgue integration →