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

We show that the principal $\beta$-Gr\"uss inequalities for the positive integral can be obtained naturally from elementary measure theory. Once the positive $\beta$-integral is recognised as integration with respect to a finite positive purely atomic measure, and this measure is normalised to a probability measure, the associated Chebyshev functional becomes simply a covariance. The corresponding inequalities then follow from standard facts valid on arbitrary probability spaces: Korkine's identity, H\"older's inequality on the product space, Cauchy's inequality for covariance, and the elementary variance bound for bounded functions. The Riemann--Stieltjes $\beta$-estimates follow, in the signed case, by domination with respect to the total variation measure. Thus, rather than adding another member to this family of $\beta$-Gr\"uss inequalities, this note identifies the elementary measure-theoretic mechanism that accounts for the family itself.