PomeDefinition 16.28.58.52560A3D59BD For $u\in \mathrm{PSH}(X,\theta )$, define
\[ \begin{split} \underline{\operatorname {vol}}_\theta (u) & =\inf _{v\in \mathrm{PSH}(X,\theta ),\ v\sim u} \int _ X\mathrm{MA}_\theta (v),\\ \overline{\operatorname {vol}}_\theta (u) & =\sup _{v\in \mathrm{PSH}(X,\theta ),\ v\sim u} \int _ X\mathrm{MA}_\theta (v). \end{split} \]
These are the lower and upper volumes of the singularity type of $u$ in the background $\theta $. When $\mathrm{PSH}(X,\theta )$ is nonempty, we write $\underline{\operatorname {vol}}(\theta )$ and $\overline{\operatorname {vol}}(\theta )$ for their values at the potential $V_\theta $ with minimal singularities.
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