Proposition 1

a) If $f$ and $g$ are two numerical functions $\geq 0$ that are equal locally almost everywhere, then $\mu^\bullet(f) = \mu^\bullet(g)$. b) If $f$ and $g$ are two numerical functions $\geq 0$ such that $f \leq g$, then $\mu^\bullet(f) \leq \mu^\bullet(g)$. c) If $f$ is a numerical function $\geq 0$, and $\alpha$ is a number $\geq 0$, then $\mu^\bullet(\alpha f) = \alpha \mu^\bullet(f)$. d) If $f$ and $g$ are two numerical functions $\geq 0$, then $\mu^\bullet(f + g) \leq \mu^\bullet(f) + \mu^\bullet(g)$. e) If $(f_n)_{n \in \mathbf{N}}$ is an increasing sequence of numerical functions $\geq 0$, and if $f = \lim_{n \to \infty} f_n$, then $\mu^\bullet(f) = \lim_{n \to \infty} \mu^\bullet(f_n)$.

The properties a), b), c), d) may be deduced immediately from the corresponding properties of the upper integral: a) from Proposition 6 of Ch. IV, §2, No. 3 and Proposition 5 of Ch. IV, §5, No. 2; b), c), d) from Propositions 10, 11, 12 of Ch. IV, §1, No. 3. To establish e), denote by $\mathfrak{K}$ the set of compact subsets of $T$; by Theorem 3 of Ch. IV, §1, No. 3, we have

Equality holds in the relation d) if $f$ and $g$ are measurable, by Cor. 4 of Th. 5, Ch. IV, §5, No. 6. More generally, we have the following result: