Measure theory and Lebesgue-like integration in two and three dimensions over the Levi-Civita field
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Abstract
In this paper, we develop the foundations for a Lebesgue-like measure and integration theory over the spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp. on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) that has similar properties to those of the Lebesgue measure of Real Analysis. Then we introduce a family of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -valued analytic functions from which we obtain a larger family of measurable functions defined on measurable subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ). We study the properties of measurable functions, we show how to integrate them over measurable subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ), and we show that the resulting integral satisfies similar properties to those of the Lebesgue integral of Real Analysis.
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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 | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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