Proposition 2

Let X be a locally compact space countable at infinity. Let $\mu$ be a positive measure on X. For all $p\in [1,+\infty [$, the space $L^p(X, \mu)$ is of countable type.

Let $(U_n)_{n\in\mathbf{N}}$ be a sequence of relatively compact open subsets of X whose union is equal to X and which satisfy $U_n\subset U_{n+1}$ for all $n\in \mathbf{N}$ (TG, I, p. 68, prop. 15). For all $n\in \mathbf{N}$, the space $\mathscr{K}(X,\overline{U}_n)$ identifies with a closed subspace of the Banach space $\mathscr{C}(U_n)$ (INT, III, p. 40, § 1, n$^o1$); since this latter space is of countable type (TG, X, p. 25, corollary), the same is true of $\mathscr{K}(X,\overline{U}_n)$ (TG, IX, p. 19, cor., (i)). Let $\mathscr{F}_n$ be a countable dense subset of $\mathscr{K}(X,\overline{U}_n)$.

Let $f\in \mathscr{L}^p(X, \mu)$ and let $\varepsilon >0$. There exists an integer $n\in \mathbf{N}$ such that $\int_{X-U_n}|f|^p< \varepsilon /2$, and there exists $g\in \mathscr{F}_n$ such that $\int_{\overline{U}_n}|f-g|^p< \varepsilon /2$. The union of the classes in $L^p(X, \mu)$ of the elements of the sets $\mathscr{F}_n$ is therefore dense in $L^p(X, \mu)$, which concludes the proof (TG, IX, p. 18, prop. 12).