Theorem 1 (countable convexity theorem)

— *Let $(f_n)$ be a sequence of functions $\geq 0$ (finite or not) defined on $X$. For $1 \leq p < +\infty$,

(5)

Set $f = \sum_{n=1}^\infty f_n$; $f$ is the upper envelope of the increasing sequence of functions $g_n = \sum_{k=1}^n f_k$; the definition of $N_p(f)$, and Th. 3 of §1, No. 3, show that $N_p(f) = \sup_n N_p(g_n)$. But $N_p(g_n) \leq \sum_{k=1}^n N_p(f_k)$ by Prop. 1, whence the inequality (5).