Proposition 3

— Let $\mu$ be a positive measure on a completely regular space $T$, and let $p \in [1, +\infty[$. The space $\mathscr{H}$ of functions $f \in \mathscr{C}^b(T)$, whose support is contained in a $\mu$-integrable open set, is dense in $\mathscr{L}^p(\mu)$.

By Prop. 15 of §1, No. 10, it suffices to show that if K is compact in T, and if $g$ is the extension to T by 0 of a function in $\mathscr{C}_+(K)$ between 0 and 1, then there exists a function $f \in \mathscr{C}_+^b(T)$, with support contained in a $\mu$-integrable open set, such that $\|f - g\|_p$ is arbitrarily small. Now, let $\varepsilon$ be a number $> 0$, U an open neighborhood of K such that $\mu^*(U - K) < \varepsilon$, V an open neighborhood of K such that $\overline{V} \subset U$, and $f$ a function with values in $[0, 1]$, continuous, equal to $g$ on K and to 0 outside V (Lemma). The function $|f - g|^p$ is then bounded above by $\varphi_{U-K}$; therefore $\|f - g\|_p \leq \varepsilon^{1/p}$, which establishes the proposition.

Remark 3). — There is an analogous statement for functions with values in a Banach space F: the subspace $\mathscr{H} \otimes F$ of $\mathscr{C}^b(T; F)$ is dense in $\mathscr{L}_F^p(\mu)$.