Measure theory and Lebesgue-like integration in two and three dimensions over the Levi-Civita field
Bibliographic record
Abstract
In this paper, we develop the foundations for a Lebesgue-like measure and integration theory over the spaces R 2 \mathcal {R}^2 and R 3 \mathcal {R}^3 , where R \mathcal {R} is the Levi-Civita field. First we review the one-dimensional theory then we extend the results to two and three dimensions. In particular, we introduce a measure on R 2 \mathcal {R}^2 (resp. on R 3 \mathcal {R}^3 ) that has similar properties to those of the Lebesgue measure of Real Analysis. Then we introduce a family of R \mathcal {R} -valued analytic functions from which we obtain a larger family of measurable functions defined on measurable subsets of R 2 \mathcal {R}^2 (resp. R 3 \mathcal {R}^3 ). We study the properties of measurable functions, we show how to integrate them over measurable subsets of R 2 \mathcal {R}^2 (resp. R 3 \mathcal {R}^3 ), and we show that the resulting integral satisfies similar properties to those of the Lebesgue integral of Real Analysis.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".