That $K\mapsto V_{K,\omega}$ is decreasing follows straightforwardly from the definition. Observe that $0\in PSH(X,\omega)$ when $\omega\geq 0$, hence $V_{X,\omega}\equiv 0$ in this case. When $\omega$ is smooth (but not positive), considering $V_{X,\omega}^{*}$ will be a useful way of constructing a positive closed current $\omega_{V_{X,\omega}^{*}}\sim\omega$ with minimal singularities (see section 4).

Assertions 2,3,4 are simple consequences of proposition 1.3. The last assertion results from the following observation: if $\varphi\in PSH(X,\omega)$ is such that $\varphi\leq 0$ on $f(K)$, then $\varphi\circ f$ belongs to $PSH(X,f^{*}\omega)$ and satisfies $\varphi\circ f\leq 0$ on $K$. ∎