A characterization of 𝜇-equicontinuity for topological dynamical systems
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Bibliographic record
Abstract
Two different notions of measure theoretical equicontinuity (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu minus"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu -</mml:annotation> </mml:semantics> </mml:math> </inline-formula>equicontinuity) for topological dynamical systems with respect to Borel probability measures appeared in works by Gilman (1987) and Huang, Lee and Ye (2011). We show that if the probability space satisfies Lebesgue’s density theorem and Vitali’s covering theorem (for example a Cantor set or a subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), then both notions are equivalent. To show this we characterize Lusin measurable maps using <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu minus"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu -</mml:annotation> </mml:semantics> </mml:math> </inline-formula>continuity points. As a corollary we also obtain a new characterization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu minus"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu -</mml:annotation> </mml:semantics> </mml:math> </inline-formula>mean equicontinuity.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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