Corollary 13.19.8.B0CCD40EE4B7 If $\varphi $ is an ordinary model of positive lower volume, then
\[ \mathcal E_{\mathrm{env}}(X,\theta ,\varphi ) =\mathcal E_{\mathrm{vol}}(X,\theta ,\varphi ) =\{ u\in \mathrm{PSH}(X,\theta ):P_\theta [u]=\varphi \} . \]
Proof.
The first equality is theorem 13.18.5. Membership in $\mathcal E_{\mathrm{env}}(X,\theta ,\varphi )$ implies $P_\theta [u]=P_\theta [\varphi ]=\varphi $ by proposition 16.28.65. Conversely, if $P_\theta [u]=\varphi $, then theorem 13.19.5 puts $u$ in $\mathcal E_{\mathrm{env}}(X,\theta ,\varphi )$.
$\square$
Pome
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