Weighted Energy Classes of Plurifinely Plurisubharmonic Functions

In the paper we introduce weighted energy classes \({\mathcal {E}}_{\chi }(\varOmega )\) of plurifinely plurisubharmonic functions on a bounded \({\mathcal {F}}\)-hyperconvex domain \(\varOmega \) in \({\mathbb {C}}^n\) and prove the existence of weak solutions of the equation of complex Monge–Ampere type in this class.