PomeProposition 16.5.52.420BBF2E1362 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$ be potentials such that $\varphi \preceq \psi \preceq \phi $. Assume that $\varphi \in \mathcal{E}(X,\theta ;\phi )$ (resp. $\mathcal{E}^1(X,\theta ;\phi )$, $\mathcal{E}^{\infty }(X,\theta ;\phi )$), then so is $\psi $.
Proposition 16.5.52420BBF2E1362
Proof. We may assume that $\varphi \leq \psi $.
The case $\mathcal{E}^{\infty }(X,\theta ;\phi )$ is trivial. The case $\mathcal{E}(X,\theta ;\phi )$ follows from theorem 13.13.11. The case $\mathcal{E}^1(X,\theta ;\phi )$ follows from the characterization of $\mathcal{E}^1(X,\theta ;\phi )$ in proposition 16.5.22 and the monotonicity of $E_{\theta }^{\phi }$ proved in 16.5.14. $\square$
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