Proposition 3

For a lower semi-continuous function $f \geq 0$ on $X$ to be negligible, it is necessary and sufficient that $f$ be zero on the support of $\mu$.

If $|\mu|^*(f) = 0$ then $|\mu|(g) = 0$ for every function $g \in \mathscr{K}_+$ such that $g \leq f$; it follows (Ch. III, § 2, No. 3, Prop. 9) that $g$ is zero on the support $S$ of $\mu$; since $f$ is the upper envelope of the functions $g \in \mathscr{K}_+$ such that $g \leq f$ (§ 1, No. 1, Lemma), $f(x) = 0$ on $S$. Conversely, if $f(x) = 0$ on $S$ then $g(x) = 0$ on $S$ for every function $g \in \mathscr{K}_+$ such that $g \leq f$, therefore (Ch. III, § 2, No. 3, Prop. 8) $|\mu|(g) = 0$, which, by definition, implies that $|\mu|^*(f) = 0$.