I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't understand the proof of the following proposition.
Let $u$, $v$ be plurisubharmonic functions defined on $\Omega$, an open subset of $\mathbb{C}^n$, such that $u,v\in L^{\infty}_{\text{loc}}(\Omega)$.
Let $O$ be a fine-open subset of $\Omega$ where $u>v$. Then $(dd^{c}\max(u,v))^{n}\rvert O=(dd^{n}u)^{n}\rvert O$.
They say that if $O$ is open then it is obvious, while if $O$ is just fine open then it is shown using a decreasing approximation sequence $u_k$ of smooth functions for $u$.
Then since $O=\bigcap_{k}(u_k>v)$ and since $(dd^{c}\max(u_k,v))^{n}\rvert O_k=(dd^{n}u_k)^{n}\rvert O_k$ (this I don't understand), where $O_k=(u_k>v)$, the statement holds.
Maybe it is trivial but I'm not able to see it.