习题 6, § 1

设T为集合，$\mathscr{F}_+(T)$的$\mathscr{I}_+$子集，$\mathscr{I}_+$的M映射为$\overline{\mathbf{R}}_+$。 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 $p(f) = \inf_{g \in \mathscr{I}_+, g \geq f} M(g)$. 证明 $p$ 是 T 上的负担（cf. Ch. IV, §1, No. 3）。 如果$f$和$g$属于到$\mathscr{F}_+(T)$并且如果A和B是T的子集，则$p(\sup(f, g)) + p(\inf(f, g)) \leq p(f) + p(g)$和$p(A \cup B) + p(A \cap B) \leq p(A) + p(B)$。