The decomposition above is called the Siu decomposition of the closed positive $(p,p)$-current $T$. Since $\nu(T,Y)=0$ if $Y$ is a $p$-codimensional subvariety not contained in $E_+(T)$, it makes sense to write $\sum_k\nu(T,Y_k)[Y_k]=\sum\nu(T,Y)[Y]$, where the sum is implicitely extended over all $p$-codimensional irreducible analytic subsets $Y\subset X$.\

### Almost positive currents

A real $(1,1)$-current $T$ on a complex manifold $X$ is said to be almost positive if $T\geq\gamma$ holds for some smooth real $(1,1)$-form $\gamma$. Let $T\geq\gamma$ be a closed almost positive $(1,1)$-current. On a small enough open set $U$ with coordinates $z=(z_1,...,z_n)$, we write $T={\partial\overline{\partial}}\varphi$ where $\varphi$ is a degree $0$ current. Since $\gamma+Ci{\partial\overline{\partial}}|z|^2$ is a positive $(1,1)$-form on $U$ for $C>0$ big enough, we get that $i{\partial\overline{\partial}}(\varphi+C|z|^2)$ is positive, which means that $\varphi+C|z|^2$ is (the current associated to) a (unique) pluri-subharmonic function on $U$. A locally integrable function $\varphi$ on $X$ such that $i{\partial\overline{\partial}}\varphi$ is almost positive is called an almost pluri-subharmonic function, and is thus locally equal to a pluri-subharmonic function modulo a smooth function.\
The Lelong number $\nu(T,x)$ of a closed almost positive $(1,1)$-current $T$ can be defined as $\nu(T+Ci{\partial\overline{\partial}}|z|^2,x)$ as above, since this does not depend on the smooth function $C|z|^2$. Consequently, the Siu decomposition of 