monotone convergence theorem

E284671

result in measure theory theorem

The monotone convergence theorem is a fundamental result in measure theory stating that the integral of a pointwise increasing sequence of nonnegative measurable functions equals the limit of their integrals.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (1)

Label	Occurrences
monotone convergence theorem canonical	4

Statements (47)

Predicate	Object
instanceOf	result in measure theory ⓘ theorem ⓘ
allows	passing limit inside integral for monotone sequences ⓘ
alsoKnownAs	Fatou's lemma ⓘ surface form: Beppo Levi theorem
appliesTo	extended real-valued functions ⓘ nonnegative measurable functions ⓘ pointwise increasing sequence of functions ⓘ
assumes	measurable space ⓘ measure ⓘ monotone increasing sequence ⓘ nonnegative functions ⓘ sequence of measurable functions ⓘ σ-finite measure (optional but common) ⓘ
category	convergence theorem in integration theory ⓘ
conclusion	integral of pointwise limit equals limit of integrals ⓘ limit of integrals is finite or +∞ consistently with limit function ⓘ ∫ lim f_n dμ = lim ∫ f_n dμ for nonnegative increasing f_n ⓘ
contrastsWith	dominated convergence theorem ⓘ surface form: bounded convergence theorem dominated convergence theorem ⓘ
field	measure theory ⓘ probability theory ⓘ real analysis ⓘ
hasVersion	version for expectations of random variables ⓘ version for sums with counting measure ⓘ
holdsFor	complete measures ⓘ σ-algebras ⓘ
implies	Fatou lemma (in some formulations) ⓘ
importance	fundamental theorem of Lebesgue integration ⓘ key tool in modern analysis ⓘ
isSpecialCaseOf	dominated convergence theorem ⓘ surface form: Lebesgue dominated convergence theorem (with monotone domination)
isStrongerThan	Fatou's lemma ⓘ surface form: Fatou lemma in the monotone case
namedAfter	Beppo Levi ⓘ
relatesConcept	Lebesgue integration ⓘ surface form: Lebesgue integral increasing sequence of functions ⓘ measurable function ⓘ nonnegative function ⓘ pointwise convergence ⓘ
requires	countable monotonicity of measure ⓘ
typicalStatement	If 0 ≤ f_1 ≤ f_2 ≤ … and f_n → f pointwise, then ∫ f_n dμ → ∫ f dμ ⓘ
usedIn	construction of Lebesgue integral ⓘ expectation of random variables ⓘ interchanging limit and integral for nonnegative increasing sequences ⓘ Probability Theory ⓘ surface form: probability theory stochastic processes ⓘ
usedToProve	Fubini's theorem ⓘ surface form: Fubini theorem (components of proofs) dominated convergence theorem ⓘ surface form: Lebesgue dominated convergence theorem Tonelli's theorem ⓘ surface form: Tonelli theorem

Referenced by (4)

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

Lebesgue integration → characterizedBy → monotone convergence theorem ⓘ

dominated convergence theorem → comparedTo → monotone convergence theorem ⓘ

Fatou's lemma → relatedTo → monotone convergence theorem ⓘ

Tonelli's theorem → relatedConcept → monotone convergence theorem ⓘ