Question

hello everybody, i'm studying the article of Bedford-Taylor "Fine topology silov boundary..." but i don't understand the proof of the following proposition.

Let u,v plurisubharmonic function defined on $\Omega$ open subset of $\mathbb{C}^n$ such that $u,v\in L^{\infty}_{loc}(\Omega)$.

Let $O$ be a fine-open subset of $\Omega$ where $u>v$. Then $(dd^{c}\max(u,v))^{n}|O=(dd^{n}u)^{n}|O$.

They say that if $O$ is open then it is obvious. while if $O$ is just fine open then using a decreasing approximation sequence $u_k$ of smooth functions for $u$.

Then since $O=\bigcap_{k}(u_k>v)$ and since $(dd^{c}\max(u_k,v))^{n}|O_k=(dd^{n}u_k)^{n}|O_k$ (this dont'understand),

where $O_k=(u_k>v)$, the statement holds.

Maybe it is trivial but i'm not able to see it. thanks

Answer 1

yes i see it, but even on $O$ $u=\max(u,v)$ so i don't understand why he passes to an approximation sequence.

Answer 2

Hi digital: I can not give comments, hence give answer here for your answer. If O is not open, then u=max(u,v) on O does not neccessarily imply that dd^cu=dd^cmax (u,v) on a neighborhood of O, so that when restricting to O you get the equality. (If you want to do derivatives, you need to do it on an open set).