On a measure-theoretic reading of β-Grüss-type inequalities
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 Lp-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 κβ=supA Pβ(A)(1−Pβ(A))≤ 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 |νu|≤ Lμβ. This reduces the principal centred signed estimate to total variation and yields its exact fixed-grid coefficient 2κβ. 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.
Versions
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BibTeX
@misc{castillo2026measuretheoretic,
title = {{On a measure-theoretic reading of $β$-Grüss-type inequalities}},
author = {K. Castillo and Â. Macedo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.06439},
eprint = {2607.06439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.06439},
note = {CC0},
}
RIS
TY - UNPB
TI - On a measure-theoretic reading of β-Grüss-type inequalities
AU - K. Castillo
AU - Â. Macedo
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2607.06439
UR - https://arxiv.org/abs/2607.06439
AB - 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))≤ 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|≤ 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.
ER -