Exercise 6, § 1

Let T be a set, $\mathscr{I}_+$ a subset of $\mathscr{F}_+(T)$, and M a mapping of $\mathscr{I}_+$ into $\overline{\mathbf{R}}_+$. We make the following hypotheses: a) For every $f \in \mathscr{I}_+$ and every real number $t \geq 0$, the function $t \cdot f$ belongs to $\mathscr{I}_+$ and $M(t \cdot f) = t \cdot M(f)$. b) If $f$ and $g$ belong to $\mathscr{I}_+$, then so do $f + g$, $\sup(f, g)$ and $\inf(f, g)$, and one has $M(f + g) = M(f) + M(g)$. c) For every increasing sequence $(f_n)_{n \geq 1}$ of elements of $\mathscr{I}_+$, the function $f = \sup_n f_n$ belongs to $\mathscr{I}_+$, and $M(f) = \lim_{n \to \infty} M(f_n)$. For every function $f \in \mathscr{F}_+(T)$, we set $p(f) = \inf_{g \in \mathscr{I}_+, g \geq f} M(g)$. Show that $p$ is an encumbrance on T (cf. Ch. IV, §1, No. 3). If $f$ and $g$ belong to $\mathscr{F}_+(T)$ and if A and B are subsets of T, then $p(\sup(f, g)) + p(\inf(f, g)) \leq p(f) + p(g)$ and $p(A \cup B) + p(A \cap B) \leq p(A) + p(B)$.