发表机构
University of Coimbra; Universidade de Trás-os-Montes e Alto Douro（科英布拉大学; 上多罗与特拉斯-奥蒙蒂斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究从初等测度论解读β - 格鲁斯型不等式，将正β - 积分视为特定测度积分并归一化，利用概率空间标准事实得出相关不等式，确定了构成此类不等式族的测度论机制。
AI 中文摘要
我们表明，正整数的主要β - 格鲁斯不等式可自然地从初等测度论得出。一旦将正β - 积分视为关于有限正纯原子测度的积分，并将此测度归一化为概率测度，相关的切比雪夫泛函就简单地成为协方差。相应不等式由任意概率空间上的标准事实得出，如科尔金恒等式、乘积空间上的赫尔德不等式、协方差的柯西不等式以及有界函数的基本方差界。在有符号情况下，黎曼 - 斯蒂尔杰斯β估计通过关于全变差测度的控制得出。本注记确定了构成该族不等式的初等测度论机制，而非添加新的β - 格鲁斯不等式。
英文摘要
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.
CommentsAccepted version. To appear in SeMA Journal
DOI：10.1007/s40324-026-00441-y
