Corollary 1

Let $\mu$ be a measure on $X$; the space $\mathscr{E}_F(\Phi)$ is dense in each of the spaces $\mathscr{L}_F^p$ ($1 \leq p < +\infty$).

For, it follows from Prop. 19 and the criterion for convergence in mean for uniform limits of functions with compact support (§ 3, No. 3, Prop. 4) that $\mathscr{E}_F(\Phi)$ is dense, for the topology of convergence in mean of order $p$, in the closure of the space $\mathscr{K}_F$ of continuous functions with compact support, whence the corollary.