Example of regular Borel measure which is not a Radon measure [closed]

Solution 1:

In Exercise 2 of §25 in [1] the following example is given: Equip $\mathbb R$ with the Sorgenfrey topology ${\cal O}_\tau$, that is, $U\in{\cal O}_\tau$ iff for every $x\in U$ there is an $\varepsilon>0$ such that $[x,x+\varepsilon[\subset U\,.$ Then,

(a) every interval $[a,b[$ with $a<b$ is closed and open at the same time. The topology ${\cal O}_\tau$ is strictly finer than the usual topology of $\mathbb R$. In particular $(\mathbb R,{\cal O}_\tau)$ is a Hausdorff space.

(b) The Borel $\sigma$-algebra of $(\mathbb R,{\cal O}_\tau)$ is identical to the usual Borel sigma algebra of $\mathbb R\,.$

...

(f) Let $\mu$ be the measure on the Borel $\sigma$-algebra that assigns zero to every countable set and $+\infty$ to every uncountable set. This is a Borel measure that is not locally finite and neither regular from inside nor from outside.

[1] H. Bauer, Measure and Integration Theory. De Gruyter, Berlin New York 2001.