Bochner integral
E613404
The Bochner integral is a generalization of the Lebesgue integral to functions taking values in Banach spaces, widely used in functional analysis and probability theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Bochner integral canonical | 2 |
| Bochner Lp spaces | 1 |
| Radon integral | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6716262 — 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: Bochner integral Context triple: [Salomon Bochner, notableFor, Bochner integral]
-
A.
Henstock–Kurzweil integral
The Henstock–Kurzweil integral is a highly general integration theory that extends and refines the Riemann integral, capable of integrating a broader class of functions while retaining many of the intuitive properties of Riemann integration.
-
B.
Lebesgue integration
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.
-
C.
Riemann–Stieltjes integral
The Riemann–Stieltjes integral is a generalization of the Riemann integral in which integration is taken with respect to a function of bounded variation rather than just the identity function, allowing more flexible treatment of sums and distributions.
-
D.
Itô integral
The Itô integral is a fundamental stochastic integral used in probability theory and mathematical finance to rigorously define integration with respect to Brownian motion and more general semimartingales.
-
E.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
- 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: Bochner integral Target entity description: The Bochner integral is a generalization of the Lebesgue integral to functions taking values in Banach spaces, widely used in functional analysis and probability theory.
-
A.
Henstock–Kurzweil integral
The Henstock–Kurzweil integral is a highly general integration theory that extends and refines the Riemann integral, capable of integrating a broader class of functions while retaining many of the intuitive properties of Riemann integration.
-
B.
Lebesgue integration
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.
-
C.
Riemann–Stieltjes integral
The Riemann–Stieltjes integral is a generalization of the Riemann integral in which integration is taken with respect to a function of bounded variation rather than just the identity function, allowing more flexible treatment of sums and distributions.
-
D.
Itô integral
The Itô integral is a fundamental stochastic integral used in probability theory and mathematical finance to rigorously define integration with respect to Brownian motion and more general semimartingales.
-
E.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
generalization of Lebesgue integral
ⓘ
integral ⓘ mathematical concept ⓘ |
| appliesTo | Banach space–valued functions ⓘ |
| assumes | complete measure space for many theorems ⓘ |
| category | vector-valued integration theory ⓘ |
| closedUnder |
almost everywhere limits under dominated convergence
ⓘ
finite linear combinations of integrable functions ⓘ |
| codomain | Banach space ⓘ |
| compatibleWith | classical Lebesgue integral on scalar functions ⓘ |
| contrastWith |
Pettis integral in nonreflexive spaces
ⓘ
Riemann integral ⓘ |
| definitionUses |
norm in Banach space
ⓘ
simple functions ⓘ |
| domain | measure space ⓘ |
| extends |
Lebesgue integral for complex-valued functions
ⓘ
Lebesgue integral for real-valued functions ⓘ |
| field |
functional analysis
ⓘ
measure theory ⓘ probability theory ⓘ |
| generalizes |
Lebesgue integral
ⓘ
linked to:
Lebesgue integration
|
| implies | Pettis integrability ⓘ |
| integrandCondition | norm is Lebesgue integrable ⓘ |
| introducedBy | Salomon Bochner ⓘ |
| linearity | is linear in the integrand ⓘ |
| namedAfter | Salomon Bochner ⓘ |
| notation | ∫ f dμ for Banach-valued f ⓘ |
| relatedTo |
Dunford integral
ⓘ
Pettis integral ⓘ |
| requires |
Bochner measurability
ⓘ
almost separably valued functions for strong measurability ⓘ integrability of norm ⓘ separability of essential range for strong measurability ⓘ strong measurability ⓘ |
| satisfies |
Fubini theorem for Banach-valued functions
ⓘ
linked to:
Fubini's theorem
dominated convergence theorem (Bochner version) ⓘ
linked to:
dominated convergence theorem
monotone convergence theorem for nonnegative scalar norms ⓘ |
| strongerThan | Pettis integrability ⓘ |
| targetSpace |
Banach space
ⓘ
complete normed vector space ⓘ |
| timeOfIntroduction | 20th century ⓘ |
| usedIn |
evolution equations
ⓘ
operator-valued integration ⓘ partial differential equations with Banach-valued data ⓘ semigroup theory ⓘ stochastic integration in Banach spaces ⓘ vector-valued Lp spaces ⓘ |
| yields |
Bochner Lp spaces
ⓘ
linked to:
Bochner integral
|
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: Bochner integral Description of subject: The Bochner integral is a generalization of the Lebesgue integral to functions taking values in Banach spaces, widely used in functional analysis and probability theory.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.