  The main purpose of this paper is to introduce and study the notion of
plurifinely-maximal plurifinely plurisubharmonic functions, which extends the
notion of maximal plurisubharmonic functions on a Euclidean domain to a
plurifine domain of C^n in a natural way. Our main result is that a finite
plurifinely plurisubharmonic function u on a plurifine domain U satisfies (dd^c
u)^n=0 if and only if u is plurifinely-locally plurifinely-maximal outside some
pluripolar set. In particular, a finite plurifinely-maximal plurisubharmonic
function u satisfies (dd^c u)^n=0.
