Theorem 3

Let $(\mathbf{f}_n)$ be a Cauchy sequence in the space $\mathscr{L}_\mathbf{F}^p$; there exists a subsequence $(\mathbf{f}_{n_k})$ of $(\mathbf{f}_n)$ having the following properties: 1° the series with general term $N_p(\mathbf{f}_{n_{k+1}} - \mathbf{f}_{n_k})$ is convergent; 2° the series with general term $\mathbf{f}_{n_{k+1}}(x) - \mathbf{f}_{n_k}(x)$ is absolutely convergent almost everywhere; 3° if $\mathbf{f}$ is a function defined on $X$ and equal almost everywhere to the limit of the sequence $(\mathbf{f}_{n_k}(x))$, then $\mathbf{f}$ belongs to $\mathscr{L}_\mathbf{F}^p$ and the sequence $(\mathbf{f}_n)$ converges in mean of order $p$ to $\mathbf{f}$; 4° there exists a lower semi-continuous function $g \geq 0$ such that $N_p(g) < +\infty$ and such that, for every $k$, $|\mathbf{f}_{n_k}(x)| \leq g(x)$ for all $x \in X$.

As in the proof of Prop. 5 of No. 3, it suffices to define the sequence $(n_k)$ by induction, in such a way that $N_p(\mathbf{f}_{n_{k+1}} - \mathbf{f}_{n_k}) \leq 2^{-k}$; parts 2° and 3° then follow from Prop. 6 of No. 3 and the fact that $\mathscr{L}_\mathbf{F}^p$ is closed in $\mathscr{F}_\mathbf{F}^p$. On the other hand, if $h(x)$ is the sum of the series with general term $|\mathbf{f}_{n_{k+1}}(x) - \mathbf{f}_{n_k}(x)|$, Th. 1 of No. 2 shows that $N_p(h) < +\infty$; therefore, by the definition of $|\mu|^*$, there exists a lower semi-continuous function $g \geq h + |\mathbf{f}_{n_1}|$ such that $N_p(g) < +\infty$, which completes the proof.