On a measure-theoretic reading of $β$-Grüss-type inequalities
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.