Corollary 3

— Let $A$ be a countable set, $\mathfrak{F}$ a filter on $A$ having a countable base, $(f_\alpha)_{\alpha \in A}$ a family of functions $\geq 0$ in $\mathscr{L}^p$. Assume that there exists a function $g \geq 0$ such that $N_p(g) < +\infty$ and $f_\alpha \leq g$ for all $\alpha \in A$; then the function $\limsup_{\mathfrak{F}} f_\alpha$ is $p$-th power integrable and

Let $(A_n)$ be a decreasing base of $\mathfrak{F}$ and set $g_n = \sup_{\alpha \in A_n} f_\alpha$; since $A_n$ is a countable set, it follows from Cor. 2 that $g_n$ is $p$-th power integrable; on the other hand, $N_p(g_n) \geq \sup_{\alpha \in A_n} N_p(f_\alpha)$. This being so, $\limsup_{\mathfrak{F}} f_\alpha$ is the lower envelope of the decreasing sequence $(g_n)$; thus $\limsup_{\mathfrak{F}} f_\alpha$ is $p$-th power integrable by Cor. 1, and