1) If $E$ is an open subset, then $V_{E}=V_{E}^{*}$.

2) Let $E$ be a Borel subset and $P$ a $PSH(X,\omega)$-polar set. Then

3) Let $(E_{j})$ be an increasing sequence of Borel subsets and set $E=\cup E_{j}$. Then $V_{E,\omega}^{*}=\lim\searrow V_{E_{j},\omega}^{*}$ if $\omega$ is Kähler.

4) Let $K_{j}$ be a decreasing sequence of compact subsets of $X$ and set $K=\cap K_{j}$. Then $V_{K_{j},\omega}\nearrow V_{K,\omega}$, hence $V_{K_{j},\omega}^{*}\nearrow V_{K,\omega}^{*}$ a.e.

5) Fix $E\subset X$ a non-pluripolar set. Then there exists $G_{j}$ a decreasing sequence of open subsets, $E\subset G_{j}$, such that $V_{E}^{*}=\lim V_{G_{j}}^{*}$.