Proposition 19

— Let $X$ be a locally compact space, $\Phi$ a clan of subsets of $X$, containing the set of compact subsets of $X$. For every continuous mapping $f$ of $X$ into a Banach space $F$ (resp. every continuous, real-valued function $f \geqslant 0$ on $X$) with compact support $K$, there exists a sequence $(g_n)$ of functions in $\mathscr{E}_F(\Phi)$ with support contained in $K$ (resp. a sequence $(g_n)$ of functions in $\mathscr{E}(\Phi)$ such that $0 \leqslant g_n \leqslant f$ for every $n$) that converges uniformly to $f$ *(resp. $f$).

Since $f$ is uniformly continuous on $K$, one can cover $K$ by a finite number of compact sets $M_i$ ($1 \leq i \leq m$) such that the oscillation of $f$ on each $M_i$ is $\leq 1/n$. Since the $M_i$ and $K$ belong to $\Phi$, there exists a partition of $K$ into sets $N_j \in \Phi$ such that each of the sets $M_i \cap K$ is the union of a certain number of the $N_j$ (No. 9, Lemma). Let $a_j$ be an element of $F$ such that $|f(x) - a_j| \leq 1/n$ on $N_j$. Setting $g_n = \sum_j a_j \varphi_{N_j}$, we have $|f - g_n| \leq 1/n$, whence the proposition in this case. One argues similarly for a continuous real-valued function $f$, on taking $a_j = \inf_{x \in N_j} f(x)$ and $g_n = \sum_j a_j \varphi_{N_j}$.