Approximation of characteristic function of compact sets via $C_0$ functions In Kallenberg Proposition 19.14 this purely topological statement seems to be used and I don't understand it.
Let $\Omega$ be a completely regular locally compact space. Denote by $\hat{\Omega}$ the one-point compactification by the element ${*}$. Assume that we are given a Radon probability measure $\mu$ on $\hat{\Omega}$ with the property that
$$
\int_\mu \hat{u} dx=0
$$
for all $u\in C_0(\Omega)$ extended by $\hat{u}(*)=0$.
Show: All the mass lies with $*$, i.e. $\mu(*)=1$.
My approach: Let $K\subseteq\Omega$ be an arbitrary compact set. By "some" approximation theorem, be it Tietzsche or Urysohn, we find a $C_0(\Omega)$ function $u$ with $\chi_K\leq u$. Hence $\mu(K)=0$, but since the measure $\mu$ is Radon, we know that
$$
\mu(\Omega)=\sup_{K\subseteq\Omega \text{ compact}}\mu(K)=0.
$$
We conclude $\mu(*)=1-\mu(\Omega)=1$.
Question: Can somebody provide me the "some" approximation theorem with precise conclusions to this particular case? I somehow fail to find the correct argumentation here. Perhaps because the statement is false?

As mentioned in comments, the correct statement is the lemma of Urysohn, as in Rudin's "Real and complex analysis", §2.7, §2.12.
