A note on the structure of plurifinely open sets and the equality of some complex Monge-Ampère measures

Abstract

In a recent preprint published on arXiv (see arXiv:2308.02993v2, referred here as \cite{NXH}), N.X. Hong stated that every plurifinely open set $U\subset {\mathbb{C}}^n$, $n\ge 1$, is of the form $U=\bigcup \{{\phi }_j>-1\}$, where each ${\varphi }_j$ is a negative plurisubharmonic function defined on an open ball $B_j\subset {\mathbb{C}}^n$ and used this result to prove an equality result on complex Monge-Ampère measures. Unfortunately, this result is wrong as we will see below.

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