Exercise 5, § 4

Let $\mu$ be a positive measure on a locally compact space $X$. For every numerical function $f$ (finite or not), of whatever sign, defined on $X$, we denote by $\mu^*(f)$ (and call upper integral of $f$) the infimum of the numbers $\mu(h)$ for the functions $h \geq f$ that are integrable and lower semi-continuous, when such functions exist, and by $+\infty$ in the contrary case; this definition coincides with that of §1, No. 3 when $f \geq 0$. We denote by $\mu_*(f)$, and call lower integral of $f$, the number $-\mu^*(-f)$.

a) Show that if $f_1$ and $f_2$ are two numerical functions such that $f_1(x) \leq f_2(x)$ almost everywhere, then $\mu^*(f_1) \leq \mu^*(f_2)$.

b) Let $f_1$ and $f_2$ be two numerical functions such that $\mu^*(f_1) + \mu^*(f_2)$ is defined and $< +\infty$; show that $f_1(x) + f_2(x)$ is defined almost everywhere and $\mu^*(f_1 + f_2) \leq \mu^*(f_1) + \mu^*(f_2)$ (reduce to the case that $f_1(x) < +\infty$ and $f_2(x) < +\infty$ at every point, with the help of a)).

c) If $(f_n)$ is an increasing sequence of numerical functions such that $\mu^*(f_n) > -\infty$ from some index onward, show that $\mu^*(\sup_n f_n) = \sup_n \mu^*(f_n)$.