Corollary 3

Let $u$ be a complex function that is locally integrable for the positive measure $\mu$. For $u \cdot \mu$ to be a positive measure, it is necessary and sufficient that $u(t) \geqslant 0$ locally almost everywhere.

For, $u \cdot \mu$ is positive if and only if $u \cdot \mu = |u \cdot \mu| = |u| \cdot \mu$ (Prop. 2), and this is equivalent to $u = |u|$ locally almost everywhere (Cor. 2).