Proposition 4

Let $\mathscr{B}$ be a filter base on $\mathscr{F}_\mathbf{F}^p$. Assume that there exists a compact set $K \subset X$ such that, for every set $M \in \mathscr{B}$, all of the mappings $f \in M$ have their support contained in K. Under these conditions, if $\mathscr{B}$ converges uniformly in X to $f_0$, then $f_0$ belongs to $\mathscr{F}_\mathbf{F}^p$ and $\mathscr{B}$ converges in mean of order $p$ to $f_0$.

It comes to the same thing to say that, on the set of mappings $f \in \mathscr{F}_\mathbf{F}^p$ whose support is contained in a fixed compact set, the topology of uniform convergence is finer than the topology of convergence in mean of order $p$.

For, let $h$ be a continuous mapping of X into $[0,1]$, with compact support, equal to 1 on K (Ch. III, §1, No. 2, Lemma 1). For every $\varepsilon > 0$, there exists an $M \in \mathscr{B}$ such that, for every mapping $f \in M$, $|f(x) - f_0(x)| \leq \varepsilon h(x)$ for all $x \in X$. From this, it follows that $N_p(f - f_0) \leq \varepsilon N_p(h)$, whence the proposition.