A Deep Dive Into Integrals: From Riemann, Stieltjes, Lebesgue to Henstock–Kurzweil
Why this work is in the frame
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Bibliographic record
Abstract
Integration theory is broadly concerned with the existence of solutions to definite integrals over closed subintervals of the real line, and its power lies in the various techniques of integration, which enable integration of successively broader classes of functions. In this work, we establish a self–contained theory of integration in the sense of Riemann, Riemann–Stieltjes, Lebesgue, and Henstock–Kurzweil. Notably, our work utilizes the theory of Lebesgue integration developed by P. J. Daniell, which does not rely on measure theory and thus remains accessible at the undergraduate level. Our incentive for such an approach is to improve equitable access to advanced integration theory in undergraduate learning environments. This work remains grounded in the applications of integration theory, which span the disciplines of probability theory, complex analysis, quaternionic theory, and even quantum mechanics. Finally, we present the Vitali set indicator function and a novel proof that this function is not integrable in the sense of any of the integration techniques surveyed, thus crystallizing the limits of integration theory as it exists today. i
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it