In this question an example of a measure that is not Borel regular was given. However, I do not understand why this measure would not be Borel regular.
I should think that what is said only works for Borel sets which contain the given set $C$. But what about Borel sets which do not? Or does every Borel set contain $C$? How would we prove it?
Borel regularity of $\mu$ requires that for every $\mu$-measurable set $A$, there exists a Borel set $B$ containing $A$ with $\mu(A)=\mu(B)$. In the example, the set $C$ is $\mu$-measurable but any Borel set containing $C$ has strictly larger measure than $C$ itself.