Volumes of Bott-Chern classes

We study the volumes of transcendental and possibly non-closed Bott-Chern $(1,1)$-classes on an arbitrary compact complex manifold $X$. We show that the latter belongs to the class $\mathcal{C}$ of Fujiki if and only if it has the $\textit{bounded mass property}$ -- i.e., its Monge-Amp\`ere volumes have a uniform upper-bound -- and there exists a closed Bott-Chern class with positive volume. This yields a positive answer to a conjecture of Demailly-P\u{a}un-Boucksom. To this end we extend to the hermitian context the notion of non-pluripolar products of currents, allowing for the latter to be merely ${\it quasi}$-${\it closed}$ and ${\it quasi}$-${\it positive}$. We establish a quasi-monotonicity property of Monge-Amp\`ere masses, and moreover show the existence of solutions to degenerate complex Monge-Amp\`ere equations in big classes, together with uniform a priori estimates. This extends to the hermitian context fundamental results of Boucksom-Eyssidieux-Guedj-Zeriahi.