Observe that $V_{K,\omega}(x)=\sup\{\varphi(x)\,/\,\varphi\in{\mathcal{F}}_{K}\}$. Thus if ${\mathcal{F}}_{K}$ is relatively compact then it is uniformly bounded from above, hence $\sup_{X}V_{K,\omega}<+\infty$, i.e. $K$ is not $PSH(X,\omega)$-polar.

Assume conversely that $K$ is not $PSH(X,\omega)$-polar. Let $(\varphi_{j})\in{\mathcal{F}}_{K}^{\mathbb{N}}$. Then $\varphi_{j}\leq V_{K,\omega}\leq\sup_{X}V_{K,\omega}<+\infty$ hence $(\varphi_{j})$ is uniformly bounded from above. It follows from proposition 1.6 that $(\varphi_{j})$ is relatively compact. Indeed it can not converge uniformly to $-\infty$ since $\sup_{K}\varphi_{j}=0$ (see proposition 1.6). ∎