Theorem 1

— In order that a numerical function (finite or not) $f \geqslant 0$ defined on $X$ be negligible, it is necessary and sufficient that $f(x) = 0$ almost everywhere.

The condition is necessary. For, suppose that $f$ is negligible, and let $N$ be the set of $x \in X$ such that $f(x) \neq 0$; then $\varphi_N \leqslant \sup_n (n f)$, therefore $\varphi_N$ is negligible (No. 1, Props. 1 and 2).

The condition is sufficient. Suppose that the set $N$ of points where $f(x) \neq 0$ is negligible; then $f \leqslant \sup_n n \varphi_N$, therefore $f$ is negligible (No. 1, Props. 1 and 2).