{
  "abstract": "On its absolute-integrability domain, the positive $β$-integral is integration with respect to a finite positive purely atomic measure. After normalisation, its Chebyshev functional is a covariance, and its $L^p$-spaces are canonically isometric to direct sums of weighted sequence spaces. On the natural product- and square-integrability domains, the previously formulated $β$-Grüss inequalities reduce to Korkine's identity, Hölder's inequality, Cauchy-Schwarz, and elementary variance bounds. On the induced countably atomic probability space, the optimal fixed-grid coefficient is $κ_β=\\sup_A P_β(A)(1-P_β(A))\\leq 1/4$, the countably atomic counterpart of the classical finite weighted coefficient; it may be strictly smaller than $1/4$. The same reduction corrects a coefficient previously claimed to be best possible and identifies a missing sign hypothesis in a related convexity estimate. For the Riemann--Stieltjes $β$-integral, within the class of finite induced signed measures, the $β$-Lipschitz condition is equivalent to $|ν_u|\\leq Lμ_β$. This reduces the principal centred signed estimate to total variation and yields its exact fixed-grid coefficient $2κ_β$. Finally, truncation of the two atomic orbits gives positive quadrature rules with explicit tail masses. A fixed-point correction yields computable Hölder error bounds, while the uncorrected geometrically graded rule accommodates integrable singularities at the fixed point.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.06439",
  "author_total": 2,
  "authors": [
    {
      "name": "K. Castillo"
    },
    {
      "name": "Â. Macedo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9dzdq580ej7g9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.06439",
  "fulltext_url": null,
  "id": "9dzdq580ej7g9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.06439",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2607.06439",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9dzdq580ej7g9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2607.06439",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.06439"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On a measure-theoretic reading of $β$-Grüss-type inequalities",
  "title_text": "On a measure-theoretic reading of β-Grüss-type inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/9dzdq580ej7g9",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1007/s40324-026-00441-y",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}