Remark

Suppose that $\pi$ is injective. Let $\theta$ be a complex measure such that $\pi$ is $\theta$-proper and $\pi(\theta) = 0$; then $\theta = 0$. Indeed, by separating $\theta$ into its real and imaginary parts, one can reduce to the case that $\theta$ is real. We then have $\pi(\theta^+) = \pi(\theta^-)$, therefore $\theta^+ = \theta^-$ (Prop. 8), and finally $\theta = 0$.

Here is an important case where condition a) of Prop. 8 is always satisfied.