On a measure-theoretic reading of $\beta$-Gr\"uss-type inequalities

Abstract

On its absolute-integrability domain, the positive $\beta$-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 $\beta$-Gr\"uss inequalities reduce to Korkine's identity, H\"older's inequality, Cauchy--Schwarz, and elementary variance bounds. On the induced countably atomic probability space, the optimal fixed-grid coefficient is $\kappa_\beta=\sup_A P_\beta(A)(1-P_\beta(A))\leq 1/4$, 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 $\beta$-integral, within the class of finite induced signed measures, the $\beta$-Lipschitz condition is equivalent to $|\nu_u|\leq L\mu_\beta$. This reduces the principal centred signed estimate to total variation and yields its exact fixed-grid coefficient $2\kappa_\beta$. Finally, truncation of the two atomic orbits gives positive quadrature rules with explicit tail masses. A fixed-point correction yields computable H\"older error bounds, while the uncorrected geometrically graded rule accommodates integrable singularities at the fixed point.