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

We show that the principal $\beta$-Grüss 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, Hölder'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 $\beta$-Grüss inequalities, this note identifies the elementary measure-theoretic mechanism that accounts for the family itself.