Proposition 16.3.41.BE692352C556 Let $\varphi \in \mathrm{PSH}(X,\theta )$ and $f\in C^{1,\overline{1}}(X)$ with $\varphi \leq f$. Then
\[ \theta ^ n_{P_\theta [\varphi ](f)} = \mathds {1}_{\left\{ P_\theta [\varphi ](f)=f\right\} }\, \theta ^ n_ f. \]
In particular,
\[ \theta ^ n_{P_\theta [\varphi ]} = \mathds {1}_{\left\{ P_\theta [\varphi ]=0\right\} }\, \theta ^ n. \]
Proof.
By definition, we have
\[ P_\theta [\varphi ](f)=P_\theta [\varphi ]\Bigl (P_\theta (f)\Bigr ) \]
and $\varphi \leq P_\theta (f)$. By corollary 16.3.14 and corollary 13.13.61,
\[ \theta ^ n_{P_\theta [\varphi ](f)} \leq \mathds {1}_{\left\{ P_\theta [\varphi ](f)=P_\theta (f)\right\} }\, \theta ^ n_{P_\theta (f)} = \mathds {1}_{\left\{ P_\theta [\varphi ](f)=f\right\} }\, \theta ^ n_ f. \]
The reverse inequality follows immediately from theorem 13.13.57.
$\square$
Pome
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