Proposition 3

— Under the hypotheses of Prop. 2, the mapping $\alpha \mapsto \left( \langle \varphi, \alpha \rangle, \frac{\alpha}{\langle \varphi, \alpha \rangle} \right)$ is a homeomorphism of $\Gamma$ onto the product space $\mathbf{R}_+^* \times \Gamma_\varphi$.

Since the mapping $\alpha \mapsto \langle \varphi, \alpha \rangle$ is vaguely continuous, it suffices to observe that $\langle \varphi, \alpha \rangle \neq 0$ for every measure $\alpha \in \Gamma$, since $e$ belongs to the support $H_\alpha$ of $\alpha$ and $\varphi(e) > 0$.