Corollary 2

*Let $(f_n)$ be a sequence of functions in $\mathscr{L}^p$. For the upper envelope $f$ of the sequence $(f_n)$ to be p-th power integrable, it is necessary and sufficient that there exist a function $g \geqslant 0$ such that $\int^* g^p d|\mu| < +\infty$ and $f_n \leqslant g$ for all $n$.

The condition is obviously necessary, on taking $g = f^+$. Conversely, suppose it is verified, and set $g_n = \sup_{k \leqslant n} f_k$; the sequence $(g_n)$ is increasing and consists of p-th power integrable functions (No. 5, Cor. of Prop. 12). The increasing sequence of positive functions $h_n = g_n + g_1^-$ satisfies the conditions of Th. 5, since $N_p(h_n) \leqslant N_p(g + g_1^-) < +\infty$; its upper envelope $\sup_n h_n$ is therefore p-th power integrable, and the same is true of $f = \sup_n h_n - g_1^-$.