Corollary 4

— Let $\rho$ and $\theta$ be two complex measures. In order that there exist a locally $\theta$-integrable function $u$ such that $\rho = u \cdot \theta$, it is necessary and sufficient that every $\theta$-negligible compact set be $\rho$-negligible.

The condition is obviously necessary. Conversely, suppose that every $\theta$-negligible compact set is $\rho$-negligible; Theorem 2 implies the existence of a locally $|\theta|$-integrable function $g$ such that $|\rho| = g \cdot |\theta|$. On the other hand, Cor. 3 implies the existence of a function $v_1$ (resp. $v_2$), of absolute value 1 and measurable for the measure $|\rho|$ (resp. $\theta$), such that $\rho = v_1 \cdot |\rho|$ (resp. $|\theta| = \overline{v_2} \cdot \theta$). Then, by Prop. 8, $\rho = u \cdot \theta$ with $u = v_1 g \overline{v_2}$.