**6. Conclusions**

We establish several integral inequalities of Chebyshev type for continuous fields of Hermitian operators which are parametrized by an LCH space equipped with a finite Radon measure. We also obtain the Chebyshev-Grüss integral inequality via oscillations with respect to a probability Radon measure. These inequalities involve Tracy-Singh products and weighted versions of famous symmetric means. For a particular case that the LCH space is a finite space equipped with the counting measure, such integral

inequalities reduce to discrete inequalities. Our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.

**Author Contributions:** All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

**Funding:** The first author expresses his gratitude towards Thailand Research Fund for providing the Royal Golden Jubilee Ph.D. Scholarship, gran<sup>t</sup> no. PHD60K0225 to support his Ph.D. study.

**Acknowledgments:** This research was supported by King Mongkut's Institute of Technology Ladkrabang.

**Conflicts of Interest:** The authors declare no conflict of interest.
