Let \((X, d)\) be a locally compact metric space. Write \(C_{\infty}(X):=\overline{C_{c}(X)}\) for the closure of \(C_{c}(X)\) with respect to the uniform norm. Show that

(i) \(C_{\infty}(X)=\left\{u \in C(X): \forall \epsilon>0 \exists K_{\epsilon}ight.\) compact, \(\left.\sup _{x otin K_{\epsilon}}|u(x)| \leqslant \epsilonight\}\);

(ii) \(C_{\infty}(X)\) equipped with the norm \(\|\cdot\|\) is a Banach space;

(iii) for the measures \(\mu, \mu_{n} \in \mathfrak{M}_{\mathrm{r}}^{+}(X)\) we have

\[\mu_{n} \stackrel{\mathrm{v}}{ightarrow} \mu \quad \text { and } \quad \sup _{n \in \mathbb{N}} \mu_{n}(X)<\infty \Longrightarrow \int u d \mu_{n} \xrightarrow[n ightarrow \infty]{\longrightarrow} \int u d \mu \quad \forall u \in C_{\infty}(X)\]