Plurifinely Plurisubharmonic functions and the Monge Amp\`ere Operator

We will define the Monge-Amp\`ere operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets in complex n-space and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that weakly plurifinely plurisubharmonic functions are locally limits of strongly finely plurisubharmonic functions in the Dirichlet norm.