Exercise 6, § 4

a) Let $f$ be a numerical function such that $\mu^*(f)$ (Exer. 5) is finite. Show that there exists an integrable function $f_1 \geq f$ such that $\mu(f_1) = \mu^*(f)$; if $f_2$ is a second integrable function such that $f_2 \geq f$ and $\mu(f_2) = \mu^*(f)$, then $f_1$ and $f_2$ are equivalent.

b) For a numerical function $f$ to be integrable, it is necessary and sufficient that $\mu^*(f)$ and $\mu_*(f)$ be finite and equal.

c) Let $f$ be a numerical function such that $\mu^*(f)$ and $\mu_*(f)$ are both finite; let $g$ and $h$ be two integrable functions such that $g \leq f \leq h$ and $\mu(g) = \mu_*(f)$, $\mu(h) = \mu^*(f)$. Show that

d) Let $f_1, f_2$ be two numerical functions such that the numbers $\mu^*(f_1), \mu^*(f_2), \mu_*(f_1)$ and $\mu_*(f_2)$ are all finite; show that

(if $g_2$ is an integrable function such that $f_2 \leq g_2$ and $\mu^*(f_2) = \mu(g_2)$, note that, for every integrable function $h$ such that $h \leq f_1 + f_2$, one has $h - g_2 \leq f_1$). From this, deduce that if $f_1$ is integrable then

and $\mu^*(f_1 + f_2) = \mu^*(\sup(f_1, f_2)) + \mu^*(\inf(f_1, f_2))$ (for the latter relation, make use of Exer. 1 of §1).

e) Let $f$ be an integrable function. In order that a function $g$, such that $\mu^*(g)$ is finite, be integrable, it is necessary and sufficient that $\mu(f) = \mu^*(g) + \mu^*(f - g)$ (if $g_1$ is an integrable function such that $g \leq g_1$ and $\mu^*(g) = \mu(g_1)$, note that $f - g_1 \leq f - g$).