Let \((X, d)\) be a locally compact metric space and \(\mu, \mu_{n} \in \mathfrak{M}_{\mathrm{r}}^{+}(X), \mu_{n} \stackrel{\mathrm{v}}{ightarrow} \mu\). Prove that

\[\lim _{n} \int_{B} u d \mu_{n}=\int_{B} u d \mu \quad \forall u \in C_{c}(X), B \in \mathscr{B}(X), \mu(\partial B)=0\]

[ have a look at the proof of Theorem 21.15 (iii).]

Data from theorem 21.15(iii)

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(c) limn n (B) = (B) for all relatively compact Borel sets BC X such that (0B) = 0. Proof (a) (b). Let K be a compact set and use (21.6) to construct a sequence uk Cc (X) with uk + 1K. We have for all k EN lim sup 4, (K) = lim sup 1 x 44,