A note on the structure of plurifinely open sets and the equality of some complex Monge-Amp\`ere measures

TL;DR

This paper critically examines a recent claim about the structure of plurifinely open sets and its implications for complex Monge-Ampère measures, providing counterexamples and clarifications.

Contribution

It identifies errors in a recent preprint's claim about plurifinely open sets and clarifies the correct structure, impacting related measure equality results.

Findings

The claimed representation of plurifinely open sets is incorrect.

Counterexamples show the structure of such sets is more complex.

The paper clarifies the correct relationship between plurifinely open sets and 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 $U\subset \mathbb{C}^n$, $n\geq 1$, is of the form $U=\bigcup \{\varphi_j>-1\}$, where each $\phi_j$ 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-Amp\`ere measures. Unfortunately, this result is wrong as we will see below.