Let \((X, d)\) be a metric space which is separable (i.e. it contains a countable dense subset) and locally compact (i.e. every \(x \in X\) has an open neighbourhood \(U\) such that \(\bar{U}\) is compact); denote by \(\mathscr{O}\) the open sets of \(X\) and assume that \(\mu\) is a measure on \(X\) equipped with its Borel sets \(\mathscr{B}(X)=\sigma(\mathscr{O})\) such that \(\mu(K)<\infty\) for all compact sets \(K \subset X\).

(i) Show that there is a sequence of open sets \(\left(U_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{O}\) such that \(\bar{U}_{n}\) is compact and each \(U \in \mathscr{O}\) can be written as a union of sets from the sequence \(\left(U_{n}ight)_{n \in \mathbb{N}}\).

(ii) Show that \(X\) is \(\sigma\)-compact, i.e. there is a sequence of compact sets \(K_{n} \uparrow X\).

(iii) Set \(\mathcal{D}=\operatorname{span}\left\{\mathbb{1}_{U}: U=\bigcup_{n \in F} U_{n}, F \subset \mathbb{N}ight.\) finite \(\}\) and denote by \(\overline{\mathcal{D}}\) the closure of \(\mathcal{D}\) in \(\mathcal{L}^{p}(\mu), 1

[a criterion of outer regularity is given in Appendix H.]

(iv) Show that \(\mu\) is outer regular and that (ii) remains valid for \(B \in \mathscr{B}(X)\) such that \(\mu(B)<\infty\).

(v) Show that \(\overline{\mathcal{D}}=\mathcal{L}^{p}(\mu)\) and conclude that \(\mathcal{L}^{p}(\mu)\) is separable.