Exercise 1, § 1

Integration, IV

Show that if $f$ and $g$ are two numerical functions $\geqslant 0$ defined on $X$ , and $\mu$ is a positive measure on $X$ , then

\mu^*(\sup(f, g)) + \mu^*(\inf(f, g)) \leq \mu^*(f) + \mu^*(g).

From this, deduce that if $A$ and $B$ are any subsets of $X$ , then

\mu^*(A \cup B) + \mu^*(A \cap B) \leq \mu^*(A) + \mu^*(B)

(cf. §4, Exer. 8 d)).

Solution

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