*Let X be a metric space. Then every Borel measure μ on X is regular. If X is complete and separable, then the measure μ is Radon.*

This result is proved on p. 70 in  *"Measure Theory" vol. 2, Springer-Verlag, Berlin
2007,  by  V. I. Bogachev* (Theorem 7.1.7.)  An example of a regular Borel measure which is not tight is provided on the same page (Example 7.1.6).