Theorem 3.2 . [033E]

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Theorem 3.2.

Let $K$ be a Borel subset of $X$ .

1) $K$ is $PSH(X,\omega)$ -polar iff $\sup_{X}V_{K,\omega}^{*}=+\infty$ iff $V_{K,\omega}^{*}\equiv+\infty$ .

2) If $K$ is not $PSH(X,\omega)$ -polar then $V_{K,\omega}^{*}\in PSH(X,\omega)$ and satisfies $V_{K,\omega}^{*}\equiv 0$ in the interior of ${K}$ , $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $X\setminus\overline{K}$ and

\int_{\overline{K}}(\omega_{V_{K,\omega}^{*}})^{n}=\int_{X}\omega^{n}=Vol_{\omega}(X).