Proposition 8

Let $f$ be a numerical function (finite or not) defined almost everywhere; if, for every $\varepsilon > 0$, there exists two $p$-th power integrable functions $g, h$ such that $g \leq f \leq h$ almost everywhere and $N_p(h - g) \leq \varepsilon$, then $f$ is $p$-th power integrable.

For, $f$ is finite almost everywhere and $N_p(f - g) \leq N_p(h - g) \leq \varepsilon$; Prop. 7 therefore shows that $f$ is $p$-th power integrable.

Since, by definition, $\mathscr{L}_\mathbf{F}^p$ is a closed subspace of $\mathscr{F}_\mathbf{F}^p$, and since the latter space is complete (No. 3, Prop. 5), we have the following result (GT, II, §3, No. 4, Prop. 8):