Corollary 2

For a function $f$ to be locally negligible, it is necessary and sufficient that every $x \in T$ possess a neighborhood V such that $f \varphi_V$ is negligible.

For, if this property is satisfied, $f \varphi_K$ is negligible for every compact set K, and $f$ is therefore locally negligible (No. 2, Prop. 2). Conversely, suppose that $f$ is locally negligible, and let $x$ be a point of T; $x$ admits an open neighborhood V of finite measure. The function $f \varphi_V$ is then locally negligible and moderated, hence is negligible.