Proposition 16.6.10E535E4DC5384

Proof. Let

First take a candidate $\psi \in \mathrm{PSH}(X,\theta )$ in (16.6.8), so that $\psi \leq 0$ and $\psi \sim _{\mathcal{I}}\varphi $. Then $\pi ^*\psi \leq 0$ and $\pi ^*\psi \sim _{\mathcal{I}}\pi ^*\varphi $ by proposition 16.6.4; hence $\pi ^*\psi \leq \eta $. Taking the upper envelope over all such $\psi $ gives

Conversely, let $\chi \in \mathrm{PSH}(Y,\pi ^*\theta )$ be a candidate in the definition of $\eta $, so that $\chi \leq 0$ and $\chi \sim _{\mathcal{I}}\pi ^*\varphi $. By proposition 5.17.8, there is a unique $\psi \in \mathrm{PSH}(X,\theta )$ such that $\chi =\pi ^*\psi $. Since $\pi $ is surjective, $\psi \leq 0$. By proposition 16.6.4, the relation $\chi \sim _{\mathcal{I}}\pi ^*\varphi $ implies $\psi \sim _{\mathcal{I}}\varphi $. Thus $\psi \leq P_{\theta }[\varphi ]_{\mathcal{I}}$, and hence

Taking the upper envelope over all such $\chi $ gives the reverse inequality. $\square$