Triple

T2631197
Position Surface form Disambiguated ID Type / Status
Subject Lebesgue integration E59633 entity
Predicate characterizedBy P662 FINISHED
Object dominated convergence theorem
The dominated convergence theorem is a fundamental result in measure theory that provides conditions under which one can interchange limits and integrals for sequences of functions bounded by an integrable dominating function.
E284672 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: dominated convergence theorem | Statement: [Lebesgue integration, characterizedBy, dominated convergence theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: dominated convergence theorem
Context triple: [Lebesgue integration, characterizedBy, dominated convergence theorem]
  • 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. Radon–Nikodym derivative
    The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
  • D. Cauchy condensation test
    The Cauchy condensation test is a convergence criterion in mathematical analysis that determines whether an infinite series with positive, nonincreasing terms converges by comparing it to a related series formed by powers of two.
  • E. Cauchy convergence criterion
    The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: dominated convergence theorem
Triple: [Lebesgue integration, characterizedBy, dominated convergence theorem]
Generated description
The dominated convergence theorem is a fundamental result in measure theory that provides conditions under which one can interchange limits and integrals for sequences of functions bounded by an integrable dominating function.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: dominated convergence theorem
Target entity description: The dominated convergence theorem is a fundamental result in measure theory that provides conditions under which one can interchange limits and integrals for sequences of functions bounded by an integrable dominating function.
  • 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. Radon–Nikodym derivative
    The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
  • D. Cauchy condensation test
    The Cauchy condensation test is a convergence criterion in mathematical analysis that determines whether an infinite series with positive, nonincreasing terms converges by comparing it to a related series formed by powers of two.
  • E. Cauchy convergence criterion
    The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69ab4ac8596c8190b34997e73d9e991c completed March 6, 2026, 9:44 p.m.
NER Named-entity recognition batch_69abd8c567408190b94ff047efa4bc06 completed March 7, 2026, 7:50 a.m.
NED1 Entity disambiguation (via context triple) batch_69af90a7021081909f81c4ddb48fa00c completed March 10, 2026, 3:31 a.m.
NEDg Description generation batch_69af9172ba248190bbc68a00b43d9b44 completed March 10, 2026, 3:35 a.m.
NED2 Entity disambiguation (via description) batch_69af92500920819082c651f75a06dd72 completed March 10, 2026, 3:38 a.m.
Created at: March 6, 2026, 9:50 p.m.