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Let $\varphi$ be a function on a region $\Omega \in C^{n}$ which is locally integrable. Is it true that $\varphi = \psi_{1} -\psi_{2}$ where $\psi_{i}$ are psh?

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No (if I decode your abbreviation "psh" correctly, as "plurisubharmonic"). Let $n=1$. Take a function $u(x,y)=0, x<0, u(x,u)=1, x>0$. It is locally integrable, but it is not a difference of subharmonic functions. In general, subharmonic functions are sort of "almost continuous".

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Thank you. I want to change my question. What I want to know is that, given a singular hermitian metric, is it possible to characterize its singularity by the difference of two semipositive singular metric? –  Hu Zhengyu Nov 23 '12 at 16:13
    
You have to define what a "singular Hermitian metric" means. –  Alexandre Eremenko Nov 23 '12 at 20:06
    
I just know a little from Demailly's book "analytic methods in algebraic geometry". A singular hermitian metric of a line bundle is a trivialization together with a locally integrable weight. A semipositive metric is a metric with a locally psh weight. –  Hu Zhengyu Nov 23 '12 at 20:49
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