Assume $\sup_{X}V_{K,\omega}^{*}=+\infty$. By a lemma of Choquet (see lemma 4.23 in [15], chapter 1), we can find an increasing sequence of functions $\varphi_{j}\in PSH(X,\omega)$ such that $\varphi_{j}=0$ on $K$ and $V_{K,\omega}^{*}=(\lim\nearrow\varphi_{j})^{*}$. Extracting a subsequence if necessary, we can assume $\sup_{X}\varphi_{j}\geq 2^{j}$. Set $\psi_{j}=\varphi_{j}-\sup_{X}\varphi_{j}$. These functions belong to ${\mathcal{F}}_{0}$ which is a compact subfamily of $PSH(X,\omega)$ (corollary 1.7). Recall that if $\mu$ is a smooth volume form on $X$ then there exists $C_{\mu}$ such that $\int\psi_{j}d\mu\geq-C_{\mu}$ for all $j$. Set $\psi:=\sum_{j\geq 1}2^{-j}\psi_{j}$. Then $\psi\in PSH(X,\omega)$ as a decreasing limit of functions in $PSH(X,\omega)$ with $\int_{X}\psi d\mu\geq-C_{\mu}>-\infty$. Now for every $x\in K$ we get $\psi(x)=-\sum_{j\geq 1}2^{-j}\sup_{X}\varphi_{j}=-\infty$ hence $K\subset\{\psi=-\infty\}$, i.e. $K$ is $PSH(X,\omega)$-polar.

Conversely assume $K$ is $PSH(X,\omega)$-polar, $K\subset\{\psi=-\infty\}$ for some $\psi\in PSH(X,\omega)$. Then for all $c\in\mathbb{R}$, $\psi+c\in PSH(X,\omega)$ and $\psi+c\leq 0$ on $K$. Therefore $V_{K,\omega}\geq\psi+c$, $\forall c\in\mathbb{R}$. This yields $V_{K,\omega}=+\infty$ on $X\setminus\{\psi=-\infty\}$ hence $V_{K,\omega}^{*}\equiv+\infty$ on $X$ since $\{\psi=-\infty\}$ has zero volume. We have thus shown the following circle of implications: $K\text{ is }PSH(X,\omega)-\text{polar}\Rightarrow V_{K,\omega}^{*}\equiv+\infty\Rightarrow\sup_{X}V_{K,\omega}^{*}=+\infty\Rightarrow K\text{ is }PSH(X,\omega)-\text{polar}$.

Assume now that $K$ is not $PSH(X,\omega)$-polar. Then $V_{K,\omega}^{*}\in PSH(X,\omega)$ (see proposition 1.6.2) and clearly satisfies $V_{K,\omega}^{*}=0$ in the interior of ${K}$. If we show that $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $X\setminus\overline{K}$ then

as follows from Stokes theorem. Let $\varphi_{j}\in PSH(X,\omega)$ be an increasing sequence such that $\varphi_{j}=0$ on $K$ and $V_{K,\omega}^{*}=(\lim\nearrow\varphi_{j})^{*}$. Fix $B$ a small ball in $X\setminus\overline{K}$. Let $\hat{\varphi_{j}}$ be the solution of the Dirichlet problem with boundary values $\varphi_{j}$. Then $\hat{\varphi_{j}}\in PSH(X,\omega)$, $\hat{\varphi_{j}}=\varphi_{j}$ in $X\setminus B$ (in particular $\hat{\varphi_{j}}=0$ on $K$ hence $\hat{\varphi_{j}}\leq V_{K,\omega}$) and the sequence $(\hat{\varphi_{j}})$ is again increasing (theorem 2.12). Since $(\omega_{\hat{\varphi_{j}}})^{n}=0$ in $B$ and $(\lim\nearrow\hat{\varphi_{j}})=V_{K,\omega}^{*}$, it follows from the continuity of the complex Monge-Ampère on increasing sequences that $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $B$. As $B$ was an arbitrarily small ball in $X\setminus\overline{K}$ we infer $(\omega_{V_{K,\omega}^{*}})^{n}=0$ in $X\setminus\overline{K}$. ∎