Extremal ω-plurisubharmonic functions as envelopes of disc functionals

For each closed, positive (1,1)-current ω on a complex manifold X and each ω-upper semicontinuous function Φ on X we associate a disc functional and prove that its envelope is equal to the supremum of all ω-plurisubharmonic functions dominated by Φ. This is done by reducing to the case where ω has a global potential. Then the result follows from Poletsky's theorem, which is the special case ω=0. Applications of this result include a formula for the relative extremal function of an open set in X and, in some cases, a description of the ω-polynomial hull of a set.