Remark 1

Suppose that $\lambda$ is a positive measure, that $v$ is a $\lambda$-measurable function such that $|v| = 1$ locally $\lambda$-almost everywhere (which implies that $v$ is locally $\lambda$-integrable), and that $\theta = v \cdot \lambda$. Prop. 2 shows immediately that $\lambda = |\theta|$; in other words, the property of the preceding statement characterizes the positive measure $|\theta|$.