Exercise 15, § 5

Let $\mu$ be a positive measure on $X$. A numerical function $f$ (finite or not) defined on $X$ is said to be quasi-integrable if it is measurable and if $\mu^*(f) = \mu_*(f)$ ($§ 4$, Exer. 5). One then sets

again, one writes $\int f \, d\mu$ in place of $\mu(f)$.

a) Show that if $f$ and $g$ are quasi-integrable and if the sum $\mu(f) + \mu(g)$ is defined, then $f + g$ is defined almost everywhere and is quasi-integrable, and $\mu(f + g) = \mu(f) + \mu(g)$.

b) Deduce from a) that, for $f$ to be quasi-integrable, it is necessary and sufficient that $f$ be measurable and that at least one of the numbers $\mu^*(f^+), \mu^*(f^-)$ be finite; then, $\mu(f) = \mu^*(f^+) - \mu^*(f^-)$.