Proposition 6

*Let $(f_n)$ be a sequence of functions in $\mathscr{F}_\mathbf{F}^p$ such that $\sum_{n=1}^\infty N_p(f_n) < +\infty$. Under these conditions, the series with general term $f_n(x) \in \mathbf{F}$ is absolutely convergent almost everywhere in $X$. Setting $f(x) = \sum_{n=1}^\infty f_n(x)$ at the points where the series converges, and $f(x) = 0$ elsewhere, the function $f$ belongs to $\mathscr{F}_\mathbf{F}^p$ and is the sum of the series with general term $f_n$ (for the topology of convergence in mean of order $p$); more precisely, for every $n \geqslant 0$,

Consider the positive function (finite or not) $g(x) = \sum_{n=1}^\infty |f_n(x)|$. By the countable convexity theorem (No. 2, Th. 1),

thus $g$ is finite almost everywhere (§ 2, No. 3, Prop. 7), which means that the series with general term $f_n(x)$ is absolutely convergent almost everywhere. Since $\mathbf{F}$ is complete, this series is convergent almost everywhere and, for every $x \in X$, $|f(x)| \leqslant \sum_{n=1}^\infty |f_n(x)| = g(x)$, whence

which proves that $f$ belongs to $\mathscr{F}_\mathbf{F}^p$. On the other hand, for every integer $n$,

almost everywhere, whence $N_p \left( f - \sum_{k=1}^n f_k \right) \leqslant \sum_{k=n+1}^\infty N_p(f_k)$. By hypothesis, the series with general term $N_p(f_n)$ is convergent; therefore, for every $\varepsilon > 0$, there exists an integer $n$ such that $\sum_{k=n+1}^{\infty} N_p(f_k) \leq \varepsilon$, and the inequality (8) proves that $f$ is the sum of the series with general term $f_n$, for the topology of convergence in mean of order $p$.

Propositions 5 and 6 are thus completely proved.