Weighted pluricomplex energy

We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy ${\mathcal{E}}_{\chi }(\mathrm{\Omega })$ in the general case ($\chi (0)=0$ i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these classes in terms of the speed of decrease of the capacity of sublevel sets and give a complete description of the range of the operator $(d{d}^c\cdot {)}^n$ on the classes $\mathcal{E}\chi (\mathrm{\Omega })\mathrm{.}$