Corollary 2

— Let $\mathscr{E}$ be a dense subset of $\mathscr{L}_F^p$. For every function $f \in \mathscr{L}_F^p$, there exists a sequence $(g_n)$ of functions in $\mathscr{E}$ having the following properties:

1° the sequence $(g_n)$ converges in mean of order $p$ to $f$; 2° for almost every $x \in X$, the sequence $(g_n(x))$ converges to $f(x)$.

For, since the space $\mathscr{L}_F^p$ is metrizable, there exists a Cauchy sequence $(f_n)$ in $\mathscr{L}_F^p$ consisting of functions in $\mathscr{E}$ and convergent in mean of order $p$ to $f$ (GT, IX, §2, No. 6, Prop. 8); it suffices to apply Th. 3 to this sequence.

Cor. 2 is applicable in particular to the case where $\mathscr{E}$ is taken to be the space $\mathscr{K}_F$ of continuous functions with compact support.