µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces
Bibliographic record
Abstract
This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is as follows: Let X be a complete paranormed vector space and Ω an arbitrary metric space, then a sequence {μn} of probability measures is weakly convergent to a probability measure μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. A special case is as the following: if X is a Banach space, Ω an arbitrary metric space, then {μn} is weakly convergent to μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. Our theorems and corollaries in the article modified or generalized some recent results regarding the convergence of sequences of measures.
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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 | 0.006 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".