Corollary 1

In order that a function $\mathbf{f}$, with values in a Banach space or in $\overline{\mathbf{R}}$, be locally negligible for the measure $u \cdot \theta$, it is necessary and sufficient that $u \mathbf{f}$ be locally negligible for $\theta$.

To say that $\mathbf{f}$ (resp. $u \mathbf{f}$) is locally negligible for $u \cdot \theta$ (resp. for $\theta$) is equivalent to saying that $|\mathbf{f}|$ (resp. $|u| \cdot |\mathbf{f}|$) is locally negligible for $|u \cdot \theta|$ (resp. for $|\theta|$). We are thus reduced, in view of Prop. 2 of No. 2, to the case that $\mathbf{f}, u, \theta$ are positive; the statement then follows at once from Prop. 3.