Proposition 7

For a function $\mathbf{f}$ to belong to $\mathscr{L}_\mathbf{F}^p$, it is necessary and sufficient that, for every $\varepsilon > 0$, there exist a continuous function $g$ with compact support such that $N_p(\mathbf{f} - g) \leq \varepsilon$.

In other words, the functions in $\mathscr{L}_\mathbf{F}^p$ are the limits of sequences of continuous functions with compact support, for the topology of convergence in mean of order $p$.