Exercise 1, § 4

Let $H$ be a set, directed for the relation $\leqslant$, of integrable functions $\geqslant 0$ such that $\sup_{f \in H} N_1(f) < +\infty$. Let $g$ be the upper envelope of $H$; in order that $g$ be integrable and that, in $L^1$, the class $\widetilde{g}$ of $g$ be the limit of the filter of sections of the directed set of classes of the functions $f \in H$, it is necessary and sufficient that, for every $\varepsilon > 0$, there exist a function $f_1 \in H$, a set $B$, directed for $\leqslant$, of lower semi-continuous integrable functions, and a mapping $f \mapsto f^*$ of the set $H_1$ of functions $f \in H$ that are $\leqslant f_1$ into the set $B$, such that $f \leqslant f^*$ and $N_1(f^* - f) \leqslant \varepsilon$ for every function $f \in H_1$ (make use of Th. 3 of No. 4).