Fatou's lemma

E284673

lemma in measure theory result in real analysis

Fatou's lemma is a fundamental result in measure theory that provides an inequality relating the integral of the pointwise limit inferior of a sequence of nonnegative measurable functions to the limit inferior of their integrals.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (4)

Label	Occurrences
Fatou's lemma canonical	2
Beppo Levi theorem	1
Fatou lemma	1
Fatou lemma in the monotone case	1

Statements (45)

Predicate	Object
instanceOf	lemma in measure theory ⓘ result in real analysis ⓘ
appliesTo	nonnegative measurable functions ⓘ random variables as measurable functions ⓘ sequence of measurable functions ⓘ
assumption	functions are measurable ⓘ functions are nonnegative almost everywhere ⓘ measure space is fixed ⓘ
conclusion	integral of lim inf is bounded above by lim inf of integrals ⓘ
conditionOnSequence	sequence indexed by natural numbers ⓘ
contrastWith	dominated convergence theorem which gives equality under stronger assumptions ⓘ monotone convergence theorem which assumes monotone sequences ⓘ
domain	measure space ⓘ
field	measure theory ⓘ real analysis ⓘ
generalizationOf	lower semicontinuity of expectation in probability ⓘ
historicalPeriod	early 20th century ⓘ
holdsFor	extended real-valued functions ⓘ
inequalityType	lower bound inequality ⓘ
involvesConcept	Lebesgue integration ⓘ surface form: Lebesgue integral almost everywhere convergence ⓘ integral inequality ⓘ limit inferior ⓘ nonnegative functions ⓘ pointwise convergence ⓘ
languageOfOriginalPublication	French ⓘ
namedAfter	Pierre Fatou ⓘ
probabilisticForm	E[lim inf X_n] ≤ lim inf E[X_n] for nonnegative random variables ⓘ
relatedTo	Beppo Levi's lemma ⓘ dominated convergence theorem ⓘ monotone convergence theorem ⓘ
requires	σ-finite measure space (in many standard formulations) ⓘ
statementForm	∫ lim inf f_n dμ ≤ lim inf ∫ f_n dμ ⓘ
typeOf	convergence theorem ⓘ
typicalNotation	∫ lim inf_{n→∞} f_n dμ ≤ lim inf_{n→∞} ∫ f_n dμ ⓘ
usedFor	convergence theorems in probability theory ⓘ establishing lower semicontinuity of integral functionals ⓘ justifying interchange of limit and integral in one direction ⓘ proving dominated convergence theorem ⓘ proving monotone convergence theorem ⓘ
usedIn	calculus of variations ⓘ ergodic theory ⓘ functional analysis ⓘ partial differential equations ⓘ probability theory ⓘ

Referenced by (5)

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

Lebesgue integration → characterizedBy → Fatou's lemma ⓘ

monotone convergence theorem → alsoKnownAs → Fatou's lemma ⓘ

this entity surface form: Beppo Levi theorem

monotone convergence theorem → isStrongerThan → Fatou's lemma ⓘ

this entity surface form: Fatou lemma in the monotone case

dominated convergence theorem → comparedTo → Fatou's lemma ⓘ

measure theory → usesConcept → Fatou's lemma ⓘ

this entity surface form: Fatou lemma