Is every compact subset of $\Bbb{R}$ the support of some Borel measure?
I have tried to prove the exercise 2.12 in Rudin's RCA:
12 Show that every compact subset of $\Bbb{R}$ is the support of a Borel measure.
For perfect (i.e. no isolated point) compact $K$ with non-zero Lebesgue measure, we just take the Lebesgue measure. However I struck on the general case.
My attempt is as follows: For every compact $K$, we can find a decreasing sequence of open set $\langle W_n\rangle_n$ s.t. $\bigcap_n W_n =K$ and $m(W_n-K)<1/n$. Find a continuous function $h_n$ s.t.
$h_n(x) = 1$ for $x\in K$ and
$h_n(x) = 0$ outside of $V_n$.
(such function exists by Urysohn's lemma.) For continuous function $f$ with compact support define $$\Lambda f = \lim_{n\to\infty}\frac{\int_\Bbb{R} fh_ndm}{\int_\Bbb{R} h_ndm}.$$
I can not certain that the $\Lambda$ is well-defined. But if it is well-defined, then it gives an regular Borel measure (by Riesz representation theorem) $\mu$ s.t. $\mu(K)=1$. I guess that the $\mu$ has a support $K$ but I don't know how to get it. Thanks for any help!
Hint: try a sum of point masses for a countable dense subset.