Example of a Borel measure, which is not Borel-regular

Question

I have asked a question to find four types of outer measures here, and I could find three of the four examples.

We call an outer measure $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ Borel, if every Borel set $B \subset \mathbb R^n$ is $\mu$-measurable. We say, that an outer measure $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ is Borel-regular, if $\mu$ is Borel and for any subset $A \subset \mathbb R^n$ there is a Borel set $B \supset A$, such that $\mu(A) = \mu(B)$.

I would like to give an example of a Borel measure, which is not Borel-regular. Can you help me?

Answer 1

Take a set $C\subset\mathbb{R}^n$ that is not Borel. For every $X\subset \mathbb{R}^n$, let $$ \mu(X) = \begin{cases} 0 & \text{if } X\subset C; \\ \infty & \text{if } X\not\subset C. \\ \end{cases} $$ This is a measure on the entire $P(\mathbb{R}^n)$. In particular, all Borel sets are measuable.

For every Borel set $B\supset C$ we have $\mu(B)=\infty> 0=\mu(C)$, so $\mu$ is not Borel-regular.