Fatou's lemma
E284673
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.
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 |
How this entity was disambiguated
This entity first appeared as the object of triple T2631198 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Fatou's lemma Context triple: [Lebesgue integration, characterizedBy, Fatou's lemma]
-
A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
Jensen inequality
Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
-
D.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
-
E.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Fatou's lemma Target entity description: 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.
-
A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
Jensen inequality
Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
-
D.
Hölder inequality
Hölder inequality is a fundamental result in mathematical analysis that generalizes the Cauchy–Schwarz inequality and provides bounds for integrals or sums of products in Lᵖ spaces.
-
E.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Fatou's lemma Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Beppo Levi theorem
this entity surface form:
Fatou lemma in the monotone case
this entity surface form:
Fatou lemma