The weigthed Monge-Amp\`ere energy of quasiplurisubharmonic functions

We study degenerate complex Monge-Amp\`ere equations on a compact K\"ahler manifold $(X,\omega)$. We show that the complex Monge-Amp\`ere operator $(\omega + dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,\omega)$ of $\omega$-plurisubharmonic functions with finite weighted Monge-Amp\`ere energy. The class ${\mathcal E}(X,\omega)$ is the largest class of $\omega$-psh functions on which the Monge-Amp\`ere operator is well-defined and the comparison principle is valid. It contains several functions whose gradient is not square integrable. We give a complete description of the range of the Monge-Amp\`ere operator $(\omega +dd^c \cdot)^n$ on ${\mathcal E}(X,\omega)$, as well as on some of its subclasses. We also study uniqueness properties, extending Calabi's result to this unbounded and degenerate situation, and we give applications to complex dynamics and to the existence of singular K\"ahler-Einstein metrics.