Corollary 3

— Let $\theta$ be a complex measure; there exists a universally measurable function $v$, with $|v| = 1$, such that $\theta = v \cdot |\theta|$, $|\theta| = \overline{v} \cdot \theta$.

Write $\theta = \theta_1 - \theta_2 + i(\theta_3 - \theta_4)$, where $\theta_1 = (\Re \theta)^+$, $\theta_2 = (\Re \theta)^-$, $\theta_3 = (\Im \theta)^+$, $\theta_4 = (\Im \theta)^-$; the positive measures $\theta_i$ ($i = 1, 2, 3, 4$), being bounded above by $|\theta|$ (Ch. III, §1, No. 6, formula (17)), are measures with base $|\theta|$ by Theorem 2. It follows that there exists a locally $|\theta|$-integrable function $v$ such that $\theta = v \cdot |\theta|$. Proposition 2 then yields the relation $|\theta| = |v| \cdot |\theta|$, which implies that $|v| = 1$ locally $|\theta|$-almost everywhere (Cor. 2 of Prop. 3). Finally, by Prop. 8, $\overline{v} \cdot \theta = (v \overline{v}) \cdot |\theta| = |\theta|$. Since the function $v$ is defined only up to a locally $|\theta|$-negligible function, one can suppose that $v$ is universally measurable (§3, No. 4, Prop. 7) and that $|v| = 1$ everywhere.