PomeDefinition 13.22.1.A220D62C8412 For $u,v\in \mathrm{PSH}(X,\theta )$, define
\[ d_\theta (u,v)=2U(\max (u,v))-U(u)-U(v). \]
For $\delta >0$, put
\[ \mathcal M_\delta (\theta )= \{ u\in \mathrm{PSH}(X,\theta ):P_\theta [u]=u, \ \underline{\operatorname {vol}}_\theta (u)\geq \delta \} . \]
We call $d_\theta $ the volume comparison. A sequence is $d_\theta $-Cauchy if $d_\theta (u_ j,u_ k)\to 0$ as $j,k\to \infty $, and it converges to $u$ for $d_\theta $ if $d_\theta (u_ j,u)\to 0$.
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